Quantum Entanglement and Bellβs Theorem
Numerical Problems & Answers
1 PROBLEMS AND SOLUTIONS
1.1 PROBLEM 1
Two entangled particles are prepared in the singlet state. Alice measures the spin of her particle
along the z-axis, while Bob measures his particleβs spin along an axis that makes an angle of
30Β° with the z-axis. What is the probability that they obtain the same result?
Solution:
1. The probability of obtaining the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. In this case, π = 30Β°.
3. π(same)=sin2(30Β°/2)=sin2(15Β°)
4. sin(15Β°)β 0.2588
5. π(same)β(0.2588)2β 0.0670 or about 6.70%
1.2 PROBLEM 2
In a Bell test experiment, Alice and Bob each have three measurement settings (a, b, c). The
following correlation values are obtained:
πΈ(π, π)= β0.5
πΈ(π, π)= β0.7
πΈ(π, π)= β0.6
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, π)| +|πΈ(π, π)+ πΈ(πβ²,π)|, where πβ² is
the complement of b.
2. In this case, we can use π as πβ² since we only have three settings.
3. π = |πΈ(π, π)β πΈ(π, π)| +|πΈ(π, π)+ πΈ(π, π)|
4. π = |β0.5 β (β0.7)| +|β0.6 + (β0.6)|
5. π = |0.2|+|β1.2|= 0.2 + 1.2 = 1.4
1.3 PROBLEM 3
Alice and Bob share an ensemble of entangled particle pairs. Alice measures her particles along
directions π and πβ², while Bob measures along π
σ°

and π
σ°

β². The angles between these vectors are:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |β0.9239 β (β0.3827)| +|β0.3827 + 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.4 PROBLEM 4
In a Bell test experiment, the following probabilities are measured:
π(+1, +1|π, π)= 0.4
π(β1, β1|π, π)= 0.4
π(+1, β1|π, π)= 0.1
π(β1, +1|π, π)= 0.1
Calculate the correlation E(a,b).
Solution:
1. The correlation E(a,b) is given by:
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
πΈ(π, π)= π(+1, +1|π, π)+ π(β1, β1|π, π)β π(+1, β1|π, π)β π(β1, +1|π, π)
2. Substituting the given probabilities:
πΈ(π, π)= 0.4 + 0.4 β 0.1 β 0.1
3. πΈ(π, π)= 0.8 β 0.2 = 0.6
1.5 PROBLEM 5
Two entangled qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{3}}(\ket{00} + \ket{01} + \ket{10})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{11}$ component.
3. Therefore, the probability of measuring both qubits in state $\ket{1}$ is 0.
1.6 PROBLEM 6
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along three different axes: π, π
σ°

, and π, where:
β (π,π
σ°

)=120Β°
β (π
σ°

,π)=120Β°
β (π, π)=120Β°
Calculate the quantum mechanical prediction for the sum of correlations πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+
πΈ(π, π).
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. For each pair of directions:
πΈ(π,π
σ°

)= πΈ(π
σ°

, π)= πΈ(π, π)= βcos(120Β°)
3. cos(120Β°)= β0.5
4. So, πΈ(π,π
σ°

)= πΈ(π
σ°

,π)= πΈ(π, π)= β(β0.5)= 0.5
5. The sum of correlations is:
πΈ(π,π
σ°

)+ πΈ(π
σ°

,π)+ πΈ(π, π)= 0.5 + 0.5 + 0.5 = 1.5
1.7 PROBLEM 7
In a Bell test experiment, Alice and Bob share 1000 entangled particle pairs. They measure their
particlesβ spins along axes that are 60Β° apart. How many times do they expect to get the same
result (both +1 or both -1)?
Solution:
1. The probability of getting the same result is given by π(same)=sin2(π/2), where π is
the angle between the measurement axes.
2. π(same)=sin2(60Β°/2)=sin2(30Β°)= 0.25
3. The expected number of same results is:
π(same)=1000 Γ 0.25 =250
1.8 PROBLEM 8
Two qubits are prepared in the Bell state:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$$
Alice measures her qubit in the basis $\{\ket{+}, \ket{-}\}$, where $\ket{+} =
\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ and $\ket{-} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})$.
What is the probability that Bobβs qubit will be in the state $\ket{+}$ if he measures in the
same basis?
Solution:
1. First, rewrite the Bell state in the $\{\ket{+}, \ket{-}\}$ basis:
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{++} + \ket{--})$$
2. If Alice measures $\ket{+}$, Bobβs qubit will be in the state $\ket{+}$.
3. If Alice measures $\ket{-}$, Bobβs qubit will be in the state $\ket{-}$.
4. The probability of Alice measuring $\ket{+}$ is 0.5.
5. Therefore, the probability of Bobβs qubit being in the state $\ket{+}$ is also 0.5.
1.9 PROBLEM 9
In a Bell test experiment, Alice and Bob measure their particles along axes π and π
σ°

,
respectively. The angle between π and π
σ°

is 45Β°. What is the quantum mechanical prediction for
the correlation E(a,b)?
Solution:
1. The quantum correlation for two particles is given by πΈ(π,π
σ°

)= βcos(π), where π is the
angle between the measurement directions.
2. In this case, π = 45Β°.
3. πΈ(π, π
σ°

)= βcos(45Β°)
4. cos(45Β°)=1
β2β 0.7071
5. πΈ(π, π
σ°

)= β0.7071
1.10 PROBLEM 10
Alice and Bob share 10,000 entangled particle pairs. They measure their particlesβ spins along
axes that are 30Β° apart. In how many cases do they expect to get opposite results (one +1 and
one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(30Β°/2)= cos2(15Β°)β 0.9330
3. The expected number of opposite results is:
π(opposite)=10,000 Γ 0.9330 = 9,330
1.11 PROBLEM 11
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{5}}(2\ket{00} + \ket{11})$$
What is the probability of measuring the first qubit in state $\ket{0}$ and the second qubit in
state $\ket{1}$?
Solution:
1. The probability of measuring $\ket{01}$ is given by the square of the amplitude of the
$\ket{01}$ component in the state $\ket{\psi}$.
2. In the given state, there is no $\ket{01}$ component.
3. Therefore, the probability of measuring the first qubit in state $\ket{0}$ and the second
qubit in state $\ket{1}$ is 0.
1.12 PROBLEM 12
In a Bell test experiment, the following correlations are measured:
πΈ(π, π)= β0.6
πΈ(π, πβ²)= β0.8
πΈ(πβ²,π)= β0.7
πΈ(πβ², πβ²)= 0.5
Calculate the value of the CHSH inequality parameter S.
Solution:
1. The CHSH inequality is given by π = |πΈ(π, π)β πΈ(π, πβ²)| +|πΈ(πβ², π)+ πΈ(πβ²,πβ²)|
2. Substituting the given correlations:
π = |(β0.6)β(β0.8)| +|(β0.7)+ 0.5|
3. π = |0.2|+|β0.2|= 0.2 + 0.2 = 0.4
1.13 PROBLEM 13
Alice and Bob share a large number of entangled particle pairs. They measure their particlesβ
spins along four different axes: π, πβ², π
σ°

, and π
σ°

β², where:
β (π,π
σ°

)=22.5Β°
β (π, π
σ°

β²)=67.5Β°
β (πβ²,π
σ°

)=67.5Β°
β (πβ², π
σ°

β²)=112.5Β°
Calculate the quantum mechanical prediction for the CHSH inequality parameter S.
Solution:
1. The quantum correlation for two particles is given by πΈ(π₯, π¦ξ¬¦)= βcos(π), where π is the
angle between the measurement directions.
2. Calculate each correlation:
πΈ(π,π
σ°

)= βcos(22.5Β°)β β0.9239
πΈ(π,π
σ°

β²)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

)= βcos(67.5Β°)β β0.3827
πΈ(πβ²,π
σ°

β²)= βcos(112.5Β°)β 0.3827
3. The CHSH parameter is given by π = |πΈ(π, π
σ°

)β πΈ(π,π
σ°

β²)| +|πΈ(πβ², π
σ°

)+ πΈ(πβ²,π
σ°

β²)|
4. π = |(β0.9239)β(β0.3827)| +|(β0.3827)+ 0.3827|
5. π = |β0.5412|+|0|= 0.5412 + 0 = 0.5412
6. π = 2β2 β 2.8284
1.14 PROBLEM 14
Two qubits are prepared in the state:
$$\ket{\psi} = \frac{1}{\sqrt{10}}(2\ket{00} + \ket{01} + 2\ket{10} + \ket{11})$$
What is the probability of measuring both qubits in the state $\ket{1}$?
Solution:
1. The probability of measuring both qubits in state $\ket{1}$ is given by the square of the
amplitude of the $\ket{11}$ component in the state $\ket{\psi}$.
2. In the given state, the amplitude of $\ket{11}$ is 1
β10.
3. Therefore, the probability is:
$$P(\ket{11}) = \left(\frac{1}{\sqrt{10}}\right)^2 = \frac{1}{10} = 0.1$$
4. The probability of measuring both qubits in state $\ket{1}$ is 0.1 or 10%.
1.15 PROBLEM 15
In a Bell test experiment, Alice and Bob share 5000 entangled particle pairs. They measure their
particlesβ spins along axes that are 45Β° apart. How many times do they expect to get opposite
results (one +1 and one -1)?
Solution:
1. The probability of getting opposite results is given by π(opposite)= cos2(π/2), where π
is the angle between the measurement axes.
2. π(opposite)= cos2(45Β°/2)= cos2(22.5Β°)β 0.8536
3. The expected number of opposite results is:
π(opposite)=5000 Γ 0.8536 =4268
4. They expect to get opposite results approximately 4268 times.
