QUANTUM KEY DISTRIBUTION (QKD) PROTOCOLS
- BB84 AND E91
1 BB84 PROTOCOL
Problem 1. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis
reconciliation, they find that they used the same basis for 520 qubits. During the error
estimation phase, they compare 100 bits and find 3 errors. What is the estimated quantum bit
error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
2 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 9. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 10. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 11. In a BB84 implementation, Alice and Bob perform basis reconciliation and find
that they used the same basis for 400 out of 1000 qubits. What is the probability that they used
the same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
3 E91 PROTOCOL
Problem 12. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 13. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 14. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 15. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 16. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 17. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
4 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
5 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
6 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
7 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
8 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
9 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
10 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
11 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
12 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
13 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
14 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
15 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
16 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
17 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
18 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
19 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
20 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
21 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
22 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
23 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
24 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
25 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
26 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
27 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
28 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
29 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
30 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
31 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
32 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
33 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
34 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
35 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
36 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
37 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
38 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
39 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
40 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
41 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
42 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
43 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
44 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
45 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
46 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
47 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
48 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
49 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
50 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
. In a BB84 QKD implementation, Alice sends 1000 qubits to Bob. After basis reconciliation,
they find that they used the same basis for 520 qubits. During the error estimation phase, they
compare 100 bits and find 3 errors. What is the estimated quantum bit error rate (QBER)?
Solution 1. Step 1: Calculate the error rate in the sample. Error rate = Number of errors /
Sample size Error rate = 3 / 100 = 0.03 or 3%
Step 2: The estimated QBER is equal to the error rate in the sample. QBER = 3% = 0.03
Therefore, the estimated quantum bit error rate (QBER) is 3% or 0.03.
Problem 2. In a BB84 protocol implementation, Alice and Bob use two bases: rectilinear (+)
and diagonal (×). If Alice sends the following 8 qubits in the given bases, and Bob measures
them in the bases shown, what will be their shared key after basis reconciliation?
Alice’s qubits: 1 0 1 1 0 0 1 0 Alice’s bases: + × + × + × + × Bob’s bases: + + × × + + × ×
Solution 2. Step 1: Identify matching bases. Matching bases occur in positions 1, 4, 5, and 8.
Step 2: Keep only the bits where bases match. Shared key: 1 1 0 0
Therefore, after basis reconciliation, Alice and Bob’s shared key is 1100.
Problem 3. In a BB84 QKD system, the probability of a photon being detected by Bob’s
detector is 0.1. If Alice sends 10,000 photons, what is the expected number of photons that
Bob will detect?
Solution 3. Step 1: Use the expected value formula. Expected number = Number of photons
× Probability of detection Expected number = 10,000 × 0.1 = 1,000
Therefore, Bob is expected to detect 1,000 photons.
Problem 4. In a BB84 implementation, Alice and Bob perform basis reconciliation and find that
they used the same basis for 400 out of 1000 qubits. What is the probability that they used the
same basis for a given qubit?
Solution 4. Step 1: Calculate the probability. Probability = Number of matching bases / Total
number of qubits Probability = 400 / 1000 = 0.4 or 40%
Therefore, the probability that Alice and Bob used the same basis for a given qubit is 0.4 or
40%.
51 E91 PROTOCOL
Problem 5. In an E91 protocol implementation, Alice and Bob each have three measurement
angles: 0°, 45°, and 90°. If they perform 1000 measurements, what is the expected number of
times they will have matching bases?
Solution 5. Step 1: Calculate the probability of matching bases. There are 9 possible
combinations of measurement angles, and 3 of them match (0°-0°, 45°-45°, 90°-90°).
Probability of matching bases = 3 / 9 = 1 / 3
Step 2: Calculate the expected number of matching bases. Expected number = Total
measurements × Probability of matching bases Expected number = 1000 × (1 / 3) ≈ 333.33
Therefore, Alice and Bob are expected to have matching bases approximately 333 times out of
1000 measurements.
Problem 6. In an E91 experiment, Alice and Bob measure 1000 entangled pairs. They find
that for the pairs where they chose the same measurement basis, their results were correlated
98% of the time. What is the estimated quantum bit error rate (QBER)?
Solution 6. Step 1: Calculate the error rate. Error rate = 1 - Correlation rate Error rate = 1 -
0.98 = 0.02 or 2%
Step 2: The estimated QBER is equal to the error rate. QBER = 2% = 0.02
Therefore, the estimated quantum bit error rate (QBER) is 2% or 0.02.
Problem 7. In an E91 protocol, Alice and Bob measure the spin of entangled particles along
three axes: a₁, a₂, and a₃ for Alice, and b₁, b₂, and b₃ for Bob. The angles between these axes
are:
∠(a₁,b₁)=45°
∠(a₂,b₂)=45°
∠(a₃,b₃)=0°
Calculate the expected value of the CHSH (Clauser-Horne-Shimony-Holt)
inequality for this configuration.
Solution 7. Step 1: Recall the CHSH inequality formula. S = |E(a₁, b₁) - E(a₁, b₃)| + |E(a₃,
b₁) + E(a₃, b₃)|
Step 2: Calculate the expected values using E(a, b) = -cos(θ), where θ is the angle between
axes. E(a₁, b₁) = E(a₂, b₂) = -cos(45°) = -1/√2 ≈ -0.707 E(a₃, b₃) = -cos(0°) = -1 E(a₁, b₃) =
E(a₃, b₁) = -cos(45°) = -1/√2 ≈ -0.707
Step 3: Substitute these values into the CHSH inequality formula. S = |(-1/√2) - (-1/√2)| + |(-
1/√2) + (-1)| = |0| + |-1/√2 - 1| = 0 + |-(1/√2 + 1)| = 1/√2 + 1 ≈ 0.707 + 1 ≈ 1.707
Therefore, the expected value of the CHSH inequality for this configuration is approximately
1.707.
Problem 8. In an E91 protocol implementation, Alice and Bob measure 10,000 entangled
pairs. They use their matching bases measurements to generate a key and their non-matching
bases measurements to test for eavesdropping. If the probability of choosing the same basis is
1/3, what is the expected length of their generated key?
Solution 8. Step 1: Calculate the expected number of matching basis measurements.
Expected matching = Total pairs × Probability of matching bases Expected matching = 10,000
× (1/3) ≈ 3,333.33
Step 2: The key length is equal to the number of matching basis measurements. Expected key
length ≈ 3,333 bits
Therefore, the expected length of Alice and Bob’s generated key is approximately 3,333 bits.
