Chapter 39 Quiz: Bell Tests, Entanglement, and Reality

Instructions: This quiz covers the core concepts from Chapter 39. For multiple choice, select the single best answer. For true/false, provide a brief justification (1-2 sentences). For short answer, aim for 3-5 sentences. For applied scenarios, show your work.

Multiple Choice (10 questions)

Q1. The CHSH inequality states that for any local hidden variable theory:

(a) $|S| \leq 1$ (b) $|S| \leq 2$ (c) $|S| \leq 2\sqrt{2}$ (d) $|S| \leq 4$

Q2. The quantum mechanical prediction for the CHSH parameter $S$ with the singlet state and optimal measurement settings is:

(a) $|S| = 2$ (b) $|S| = 2\sqrt{2} \approx 2.828$ (c) $|S| = 3$ (d) $|S| = 4$

Q3. The Tsirelson bound states that for any quantum state and measurements:

(a) $|S| \leq 2$ (b) $|S| \leq 2\sqrt{2}$ (c) $|S| \leq 3$ (d) $|S| \leq 4$

Q4. The quantum correlation function for the singlet state $|\Psi^-\rangle$ with measurement axes separated by angle $\theta$ is:

(a) $E = \cos\theta$ (b) $E = -\cos\theta$ (c) $E = \sin\theta$ (d) $E = -\sin\theta$

Q5. The detection loophole in Bell tests refers to the possibility that:

(a) Detectors might not obey the laws of quantum mechanics (b) Undetected particles could have different correlations than detected ones, allowing a local model to fake violations (c) The detectors are too slow to capture the photons (d) Dark counts in the detectors create false correlations

Q6. The 2015 Delft loophole-free Bell test used:

(a) Entangled photons from SPDC, separated by 400 km (b) Nitrogen-vacancy centers in diamond, separated by 1.3 km (c) Trapped ions, separated by 10 m (d) Superconducting qubits on a single chip

Q7. In the E91 quantum key distribution protocol, security is guaranteed by:

(a) The computational difficulty of factoring large numbers (b) The no-cloning theorem (c) The observed Bell inequality violation, which bounds the eavesdropper's information (d) Classical one-way functions

Q8. Device-independent QKD differs from standard QKD in that:

(a) It does not require any quantum devices (b) Security is guaranteed by observed statistics alone, with no assumptions about the devices' internal workings (c) It uses classical channels instead of quantum channels (d) It requires trusted third-party devices

Q9. A Werner state $\hat{\rho} = V|\Psi^-\rangle\langle\Psi^-| + \frac{1-V}{4}\hat{I}_4$ violates the CHSH inequality when:

(a) $V > 0$ (b) $V > 1/3$ (c) $V > 1/\sqrt{2} \approx 0.707$ (d) $V = 1$

Q10. Bell's theorem, combined with experimental results, proves that:

(a) Faster-than-light communication is possible (b) The Copenhagen interpretation is the correct interpretation of quantum mechanics (c) Nature is not simultaneously local and realistic (d) Hidden variable theories are impossible

True/False (4 questions)

Q11. True or False: A Bell inequality violation allows Alice to send information to Bob faster than light.

Q12. True or False: An entangled quantum state always violates a Bell inequality.

Q13. True or False: The freedom-of-choice loophole can be completely closed by using quantum random number generators for setting choices.

Q14. True or False: The minimum detection efficiency required to close the detection loophole for the CHSH inequality with the singlet state is approximately 82.8%.

Short Answer (4 questions)

Q15. Explain the physical meaning of "spacelike separation" in the context of a Bell test. Why is spacelike separation between the measurement events necessary for closing the locality loophole?

Q16. Describe the BB84 quantum key distribution protocol in 3-5 sentences. What is the role of the quantum channel, and what information is communicated over the classical channel?

Q17. What is self-testing, and how does it relate to Bell inequality violations? Give one practical application.

Q18. Explain why a Werner state with visibility $V = 0.5$ is entangled (it cannot be written as a mixture of product states) but does not violate the CHSH inequality. What does this tell us about the relationship between entanglement and nonlocality?

Applied Scenarios (2 questions)

Q19. You are running a Bell test with entangled photon pairs. Your source produces pairs at a rate of $50{,}000$ per second. Your detector efficiency is $\eta = 92\%$. Your state visibility is $V = 95\%$. The measurement settings are optimal for CHSH.

(a) What is the expected coincidence rate (pairs where both photons are detected)?

(b) What is the expected CHSH parameter $S$?

(c) What is the approximate standard error $\sigma_S$ after $T = 10$ seconds of data collection?

(d) What is the statistical significance (number of $\sigma$) of the Bell violation?

(e) Is the detection loophole closed in this experiment? Justify your answer.

Q20. Alice and Bob are implementing the E91 quantum key distribution protocol. They collect $N = 100{,}000$ entangled pairs. One-third of the setting combinations are used for the CHSH test, and the remaining (those where Alice and Bob chose the same effective basis) are used for key generation.

(a) Approximately how many pairs contribute to the CHSH test?

(b) If the observed CHSH value is $S = 2.75 \pm 0.02$, compute Eve's maximum information per bit using the formula $I_{\text{Eve}} \leq h\left(\frac{1 + \sqrt{(S/2)^2 - 1}}{2}\right)$, where $h(x) = -x\log_2 x - (1-x)\log_2(1-x)$ is the binary entropy.

(c) Approximately how many raw key bits are generated from the matched-basis pairs?

(d) After privacy amplification (which removes Eve's information), approximately how many secure key bits remain?

Answer Key

Q1: (b) $|S| \leq 2$

Q2: (b) $|S| = 2\sqrt{2} \approx 2.828$

Q3: (b) $|S| \leq 2\sqrt{2}$

Q4: (b) $E = -\cos\theta$

Q5: (b) Undetected particles could have different correlations than detected ones, allowing a local model to fake violations

Q6: (b) Nitrogen-vacancy centers in diamond, separated by 1.3 km

Q7: (c) The observed Bell inequality violation, which bounds the eavesdropper's information

Q8: (b) Security is guaranteed by observed statistics alone, with no assumptions about the devices' internal workings

Q9: (c) $V > 1/\sqrt{2} \approx 0.707$

Q10: (c) Nature is not simultaneously local and realistic

Q11: False. Bell violations reveal correlations visible only when data is compared after the fact via a classical channel. The no-signaling theorem guarantees that Alice's marginal probabilities are independent of Bob's setting choice, so no information can be transmitted.

Q12: False. There exist entangled states that do not violate any Bell inequality. The Werner state with $1/3 < V < 1/\sqrt{2}$ is entangled but admits a local hidden variable model for the CHSH inequality. More generally, entanglement is necessary but not sufficient for Bell violation.

Q13: False. Quantum random number generators make the freedom-of-choice loophole extremely implausible, but they cannot logically rule out superdeterminism — the possibility that the "random" outputs are correlated with the hidden variables due to a common cause in the distant past. No experiment can fully close this loophole.

Q14: True. Eberhard showed that for the CHSH inequality with the singlet state, the threshold is $\eta_{\min} = 2/(1 + \sqrt{2}) \approx 82.8\%$. Below this efficiency, a local hidden variable model can reproduce the observed statistics by exploiting the detection loophole.

Q15: Two events are spacelike separated if neither lies in the future light cone of the other — no signal traveling at or below the speed of light can connect them. In a Bell test, spacelike separation between Alice's measurement event (setting choice + outcome) and Bob's measurement event ensures that neither party's result can be influenced by the other's setting. Without this, a local model could explain violations: information about Alice's setting could travel to Bob's detector and influence his outcome.

Q16: In BB84, Alice encodes random bits into qubits using randomly chosen bases ($Z$ or $X$) and sends them to Bob through a quantum channel. Bob measures each qubit in a randomly chosen basis. After all measurements, they publicly compare their basis choices (but not outcomes) over a classical channel, keeping only the bits where they chose the same basis (the sifted key). They then sacrifice a fraction of the sifted key to estimate the error rate; if it exceeds a threshold (~11%), they abort (eavesdropper detected). The quantum channel carries the qubits; the classical channel carries basis choices and error correction information.

Q17: Self-testing is the ability to characterize a quantum state and measurements based solely on the observed measurement statistics, without any assumptions about the devices. If Alice and Bob observe a maximal CHSH violation ($S = 2\sqrt{2}$), they can conclude that their shared state is (equivalent to) the singlet and their measurements are (equivalent to) the optimal CHSH observables, up to local isometries. One practical application is certified quantum random number generation: the outputs of a Bell test are provably random, even to the device manufacturer.

Q18: A Werner state with $V = 0.5$ is entangled because it has a negative partial transpose (violates the Peres-Horodecki criterion), confirming inseparability. However, the CHSH parameter is $S = 2\sqrt{2} \times 0.5 \approx 1.41 < 2$, so it does not violate CHSH. This demonstrates that entanglement and nonlocality (Bell violation) are distinct quantum resources. Entanglement is necessary for Bell violation, but not sufficient — some entangled states have correlations that can be reproduced by local hidden variable models for projective measurements.

Q19: (a) Coincidence rate = $50{,}000 \times (0.92)^2 = 50{,}000 \times 0.8464 = 42{,}320$ pairs/s. (b) $S = 2\sqrt{2} \times V = 2\sqrt{2} \times 0.95 \approx 2.687$. (c) Total coincidences in 10 s: $N = 423{,}200$. Standard error: $\sigma_S \approx 4/\sqrt{N} = 4/\sqrt{423{,}200} \approx 0.00615$. (d) Significance: $(|S| - 2)/\sigma_S = (2.687 - 2)/0.00615 \approx 111.7\sigma$. (e) Yes: $\eta = 92\% > 82.8\%$, so the detection loophole is closed.

Q20: (a) In E91, 4 of the 9 setting combinations are used for the CHSH test. So approximately $4/9 \times 100{,}000 \approx 44{,}444$ pairs. (b) $(S/2)^2 - 1 = (2.75/2)^2 - 1 = 1.890625 - 1 = 0.890625$. $\sqrt{0.890625} \approx 0.9438$. $(1 + 0.9438)/2 = 0.9719$. $h(0.9719) = -0.9719\log_2(0.9719) - 0.0281\log_2(0.0281) \approx 0.041 + 0.148 = 0.189$ bits. So $I_{\text{Eve}} \leq 0.189$ bits per bit. (c) The matched-basis pairs are approximately $1/9 \times 100{,}000 \approx 11{,}111$ raw key bits (the specific number depends on the exact setting geometry; approximately $11{,}000$-$22{,}000$). (d) After privacy amplification, secure key bits $\approx N_{\text{raw}} \times (1 - I_{\text{Eve}} - \text{leak}_{\text{EC}}) \approx 11{,}111 \times (1 - 0.189 - 0.1) \approx 11{,}111 \times 0.711 \approx 7{,}900$ bits. (The exact number depends on the error correction leakage.)