Chapter 24 Exercises: Entanglement, Bell's Theorem, and Foundations

Part A: Conceptual Questions (difficulty: moderate)

These questions test your understanding of the core ideas. No calculations required unless stated.

A.1 State the two key assumptions of the EPR argument (locality and realism) in your own words. For each assumption, give a concrete everyday example where the assumption seems obviously true, and explain why quantum mechanics challenges it.

A.2 A friend says: "EPR showed that quantum mechanics is wrong." Explain precisely what EPR actually showed. What did Einstein conclude, and why is your friend's summary inaccurate?

A.3 Explain why the glove analogy (Section 24.2) fails to capture quantum entanglement. What specific prediction distinguishes entangled quantum particles from pairs of gloves with pre-determined handedness? Be quantitative.

A.4 In the CHSH inequality derivation (Section 24.4), we assumed that $A(\hat{a}_1, \lambda)$ does not depend on Bob's measurement setting $\hat{b}_j$. What physical assumption does this encode? What would it mean, operationally, if this assumption failed?

A.5 Explain why the Tsirelson bound $|S| \leq 2\sqrt{2}$ is surprising from two directions: (a) why is it surprising that it exceeds 2? (b) why is it surprising that it does not reach 4?

A.6 A skeptic claims: "Aspect's experiments don't prove anything because only a small fraction of photon pairs were detected. Maybe the detected pairs were a biased sample." Is this a legitimate concern? What is it called? How was it addressed by the 2015 experiments?

A.7 In quantum teleportation, Alice destroys the original state $|\chi\rangle$ when she performs her Bell measurement. Explain why this is not merely a technical limitation but a necessary consequence of the no-cloning theorem. What would go wrong if teleportation could proceed without destroying the original?

A.8 Explain the difference between quantum teleportation and classical faxing. In particular: (a) Does teleportation transmit information faster than light? (b) Does the original survive? (c) Is the copy perfect? (d) How much classical information is needed?

A.9 Present one strength and one weakness of each of the following interpretations: Copenhagen, many-worlds, Bohmian mechanics, QBism. Do so fairly — a defender of each interpretation should consider your summary accurate.

A.10 John Bell was sympathetic to Bohmian mechanics. Given that Bohmian mechanics is explicitly non-local, explain why Bell did not consider this a fatal flaw. What did Bell consider more important than locality?

Part B: Applied Problems (difficulty: moderate to hard)

These problems require direct calculations using the chapter's key results.

B.1: Singlet Correlations

The singlet state is $|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|{\uparrow\downarrow}\rangle - |{\downarrow\uparrow}\rangle)$.

(a) Compute $E(\hat{a}, \hat{b}) = \langle \Psi^- | (\hat{\boldsymbol{\sigma}}_A \cdot \hat{a})(\hat{\boldsymbol{\sigma}}_B \cdot \hat{b}) | \Psi^- \rangle$ for $\hat{a} = \hat{z}$ and $\hat{b} = \hat{z}$. Verify that you get $-1$ (perfect anticorrelation).

(b) Compute $E(\hat{a}, \hat{b})$ for $\hat{a} = \hat{z}$ and $\hat{b} = \hat{x}$. Verify that you get $0$ (no correlation).

(c) Compute $E(\hat{a}, \hat{b})$ for general axes $\hat{a}$ and $\hat{b}$ separated by angle $\theta$. Show $E = -\cos\theta$.

(d) Plot $E(\theta)$ for $\theta \in [0, 2\pi]$. On the same plot, show the correlation function $E_{\text{LHV}}(\theta)$ for the "best" local hidden variable model: $E_{\text{LHV}}(\theta) = 1 - \frac{2|\theta|}{\pi}$ for $|\theta| \leq \pi$. Where do they differ most?

B.2: Bell Inequality Violation

(a) Using the original Bell inequality $|E(\hat{a}, \hat{b}) - E(\hat{a}, \hat{c})| \leq 1 + E(\hat{b}, \hat{c})$, find the set of three measurement angles that maximizes the quantum violation. (Hint: parametrize $\hat{a}$ at angle 0, $\hat{b}$ at angle $\theta$, $\hat{c}$ at angle $2\theta$, and optimize over $\theta$.)

(b) What is the maximum violation (the maximum value of the left side minus the right side)?

(c) Show that for $\theta = 60°$, the quantum prediction satisfies $|E(\hat{a}, \hat{b}) - E(\hat{a}, \hat{c})| = 1$ and $1 + E(\hat{b}, \hat{c}) = 0.5$, violating the inequality.

B.3: CHSH Parameter Calculation

(a) Verify the optimal CHSH angles given in Section 24.4: Alice at $0°$ and $90°$, Bob at $45°$ and $-45°$. Compute $S$ explicitly for the singlet state.

(b) Now try Alice at $0°$ and $90°$, Bob at $0°$ and $90°$. What is $S$? Does it violate the CHSH inequality? Explain.

(c) Show that for the product state $|{\uparrow\uparrow}\rangle$, $|S| \leq 2$ for any choice of measurement angles. (Hint: compute $E(\hat{a}, \hat{b})$ for a product state and show it factors.)

(d) Suppose a source produces the Werner state $\rho_W = p|\Psi^-\rangle\langle\Psi^-| + (1-p)\frac{1}{4}\hat{I}$ (a mixture of the singlet and white noise). For what values of $p$ does the Werner state violate the CHSH inequality? (Answer: $p > 1/\sqrt{2} \approx 0.707$.)

B.4: Teleportation Protocol

Alice wants to teleport $|\chi\rangle = \frac{1}{\sqrt{3}}|0\rangle + \sqrt{\frac{2}{3}}|1\rangle$ to Bob. They share the Bell pair $|\Phi^+\rangle$.

(a) Write out the full three-qubit state $|\chi\rangle_C \otimes |\Phi^+\rangle_{AB}$ in the computational basis.

(b) Rewrite this state using the Bell basis for qubits $C$ and $A$. Show that Bob's qubit is in one of four states depending on Alice's measurement outcome.

(c) For each of Alice's four possible outcomes, write Bob's state and the correction he must apply.

(d) Verify that after Bob's correction, his qubit is in the state $|\chi\rangle$ for all four outcomes.

(e) What is the probability of each of Alice's measurement outcomes?

B.5: Superdense Coding

Alice and Bob share $|\Phi^+\rangle$. Alice applies $\hat{\sigma}_y$ to her qubit.

(a) What is the resulting two-qubit state?

(b) Show that this state is $i|\Psi^-\rangle$ (up to a global phase, which is physically irrelevant).

(c) If Bob performs a Bell measurement, what outcome does he get? What 2-bit message does this encode?

(d) Prove that the four states $\{|\Phi^+\rangle, |\Phi^-\rangle, |\Psi^+\rangle, |\Psi^-\rangle\}$ are mutually orthogonal. Why is orthogonality essential for superdense coding to work?

B.6: No-Cloning

(a) Reproduce the proof of the no-cloning theorem from Section 24.9.

(b) Suppose a "cloning machine" works for the computational basis: $\hat{U}|0\rangle|0\rangle = |0\rangle|0\rangle$ and $\hat{U}|1\rangle|0\rangle = |1\rangle|1\rangle$. What does this machine do to $|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$? Is the result a clone of $|\psi\rangle$?

(c) Show that the result in (b) is the Bell state $|\Phi^+\rangle$, not the product state $|\psi\rangle \otimes |\psi\rangle$. This machine creates entanglement, not copies.

Part C: Advanced Problems (difficulty: hard)

These problems require deeper analysis, synthesis across sections, or extended calculations.

C.1: Entanglement of Bell States

(a) Compute the reduced density matrix $\rho_A = \text{Tr}_B(|\Phi^+\rangle\langle\Phi^+|)$ by performing the partial trace explicitly.

(b) Show that $\rho_A = \frac{1}{2}\hat{I}$ — the maximally mixed state. Interpret this: what does Alice learn about her particle by measuring it?

(c) Compute the von Neumann entropy $S(\rho_A)$ for all four Bell states. Show that $S = 1$ for all of them (maximum entanglement for two qubits).

(d) Now consider the state $|\psi(\theta)\rangle = \cos\theta |00\rangle + \sin\theta |11\rangle$. Compute the entanglement entropy $E(\theta)$ as a function of $\theta$. At what value of $\theta$ is the entanglement maximized? Minimized?

C.2: CHSH Optimization

(a) For the singlet state, write the CHSH parameter $S$ as a function of four angles $\alpha_1, \alpha_2, \beta_1, \beta_2$ (the polar angles of Alice's and Bob's measurement axes in the $xz$-plane).

(b) Maximize $|S|$ analytically. Show that the optimal configuration has the four measurement axes evenly spaced at $45°$ intervals. Derive $|S|_{\max} = 2\sqrt{2}$.

(c) For the state $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$, the correlation function is $E(\hat{a}, \hat{b}) = \cos(\alpha_a - \beta_b) \cos(\alpha_a + \beta_b) - \sin(\alpha_a - \beta_b)\sin(\alpha_a + \beta_b)$... Actually, compute $E(\hat{a}, \hat{b})$ directly from the state. Show that the maximum CHSH violation is the same as for the singlet state, but with different optimal angles.

C.3: Tsirelson's Bound

(a) Define the CHSH operator $\hat{S} = \hat{A}_1 \otimes \hat{B}_1 + \hat{A}_1 \otimes \hat{B}_2 + \hat{A}_2 \otimes \hat{B}_1 - \hat{A}_2 \otimes \hat{B}_2$, where $\hat{A}_i, \hat{B}_j$ are observables with eigenvalues $\pm 1$.

(b) Compute $\hat{S}^2$. Show that $\hat{S}^2 = 4\hat{I} - [\hat{A}_1, \hat{A}_2] \otimes [\hat{B}_1, \hat{B}_2]$.

(c) Use the fact that $\|[\hat{A}_1, \hat{A}_2]\| \leq 2$ and $\|[\hat{B}_1, \hat{B}_2]\| \leq 2$ (since $\hat{A}_i^2 = \hat{B}_j^2 = \hat{I}$) to show $\hat{S}^2 \leq 8\hat{I}$.

(d) Conclude that $|S| = |\langle\hat{S}\rangle| \leq \sqrt{\langle\hat{S}^2\rangle} \leq 2\sqrt{2}$. This is the Tsirelson bound.

C.4: Teleportation with Noise

Suppose Alice and Bob share a noisy Bell pair — a Werner state $\rho_W = p|\Phi^+\rangle\langle\Phi^+| + (1-p)\frac{1}{4}\hat{I}$ instead of a perfect Bell state.

(a) Show that if Alice performs the teleportation protocol with this mixed state, the output state Bob receives (after correction) is:

$$\rho_{\text{out}} = p|\chi\rangle\langle\chi| + (1-p)\frac{1}{2}\hat{I}$$

This is a depolarizing channel applied to $|\chi\rangle$.

(b) Compute the fidelity $F = \langle\chi|\rho_{\text{out}}|\chi\rangle$ as a function of $p$.

(c) What is the fidelity when $p = 1$ (perfect Bell pair)? When $p = 0$ (no entanglement)?

(d) What is the minimum value of $p$ for which teleportation does better than the best classical strategy? (Hint: the best classical strategy without entanglement has fidelity $F_{\text{classical}} = 2/3$ for qubit states.)

C.5: GHZ State and Mermin's Inequality

Consider the three-qubit GHZ state:

$$|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}\bigl(|000\rangle + |111\rangle\bigr)$$

(a) Show that this state is genuinely tripartite entangled — it cannot be written as a product of any bipartition.

(b) Consider measurements of $\hat{\sigma}_x$ and $\hat{\sigma}_y$ on each qubit. Compute $\langle \hat{\sigma}_x^{(1)} \hat{\sigma}_y^{(2)} \hat{\sigma}_y^{(3)} \rangle$, $\langle \hat{\sigma}_y^{(1)} \hat{\sigma}_x^{(2)} \hat{\sigma}_y^{(3)} \rangle$, and $\langle \hat{\sigma}_y^{(1)} \hat{\sigma}_y^{(2)} \hat{\sigma}_x^{(3)} \rangle$ for $|\text{GHZ}\rangle$.

(c) Show that a local hidden variable theory predicts $\langle \hat{\sigma}_x^{(1)} \hat{\sigma}_x^{(2)} \hat{\sigma}_x^{(3)} \rangle = -1$ given the correlations in (b), while quantum mechanics predicts $\langle \hat{\sigma}_x^{(1)} \hat{\sigma}_x^{(2)} \hat{\sigma}_x^{(3)} \rangle = +1$. This is all-or-nothing — not a statistical inequality but a deterministic contradiction.

(d) This is Mermin's version of the GHZ argument (1990). Explain why this is, in some sense, an even stronger refutation of local realism than the CHSH inequality. What is the advantage of all-or-nothing proofs over statistical inequalities?

C.6: Entanglement and Mixed States

(a) Show that the Bell state $|\Psi^-\rangle$ can be written as $|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|{+}\rangle|{-}\rangle - |{-}\rangle|{+}\rangle)$ where $|{\pm}\rangle = \frac{1}{\sqrt{2}}(|{\uparrow}\rangle \pm |{\downarrow}\rangle)$ are the $\hat{S}_x$ eigenstates. This confirms the rotational invariance of the singlet.

(b) Define the concurrence for a two-qubit density matrix $\rho$ as $C(\rho) = \max(0, \lambda_1 - \lambda_2 - \lambda_3 - \lambda_4)$ where $\lambda_i$ are the eigenvalues (in decreasing order) of $\sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}$ with $\tilde{\rho} = (\hat{\sigma}_y \otimes \hat{\sigma}_y)\rho^*(\hat{\sigma}_y \otimes \hat{\sigma}_y)$.

Compute the concurrence of the Bell state $|\Phi^+\rangle$ and verify $C = 1$.

(c) Compute the concurrence of the Werner state $\rho_W$ as a function of $p$. For what range of $p$ is the state entangled? (Answer: $p > 1/3$.)

(d) Compare the threshold for entanglement ($p > 1/3$) with the threshold for CHSH violation ($p > 1/\sqrt{2}$). Explain in words what this means: there exist entangled states that do not violate any Bell inequality. This is the Werner bound.

Part D: Computational Problems (difficulty: variable)

These problems require writing or modifying Python code. Use the examples in code/example-01-bell.py as a starting point.

D.1 Modify the LHV simulation in example-01-bell.py to try different deterministic strategies for Alice and Bob. Show that no strategy produces $|S| > 2$ over a large number of trials.

D.2 Extend the CHSH calculation to compute $S$ for the $|\Phi^+\rangle$ Bell state as a function of Alice's and Bob's measurement angles. Produce a 2D heatmap of $S(\alpha, \beta)$ where $\alpha$ is the angle between Alice's two settings and $\beta$ is the angle between Bob's two settings. Find the maximum visually and numerically.

D.3 Implement a Monte Carlo simulation of a Bell test: generate $N$ entangled pairs, randomly choose measurement settings, compute correlations, and extract $S$ with error bars. Verify that for $N = 10{,}000$ trials, the statistical error is small enough to distinguish $S = 2\sqrt{2}$ from $S = 2$ at 5-sigma significance.

D.4 Implement the quantum teleportation protocol for an arbitrary input state $|\chi\rangle = \alpha|0\rangle + \beta|1\rangle$ using the code in project-checkpoint.py. Verify that the output matches the input for 10 randomly chosen input states. Compute the fidelity $|\langle\chi|\chi_{\text{out}}\rangle|^2$ for each.

D.5 Implement a simulation of the no-cloning theorem: build a "CNOT cloner" (which clones the computational basis) and show that it fails to clone superposition states. Compute the fidelity of the "clone" as a function of the input state.

D.6 Build a Werner state CHSH violation detector: for $p$ ranging from 0 to 1, compute the maximum $|S|$ achievable with optimal measurement angles. Plot $|S|(p)$ and mark the classical bound $|S| = 2$ and the entanglement threshold $p = 1/3$.

D.7 Implement the superdense coding protocol: Alice and Bob share $|\Phi^+\rangle$. Write a function superdense_encode(message_bits, alice_qubit) that applies the correct Pauli operation, and a function superdense_decode(two_qubit_state) that performs a Bell measurement and returns the two-bit message. Verify that all four messages (00, 01, 10, 11) are correctly transmitted.

D.8 Construct the GHZ state $|\text{GHZ}\rangle = \frac{1}{\sqrt{2}}(|000\rangle + |111\rangle)$ and verify the Mermin inequality violation numerically. Compute $\langle \sigma_x \sigma_y \sigma_y \rangle$, $\langle \sigma_y \sigma_x \sigma_y \rangle$, $\langle \sigma_y \sigma_y \sigma_x \rangle$, and $\langle \sigma_x \sigma_x \sigma_x \rangle$ for the GHZ state. Confirm that the quantum predictions contradict the local hidden variable prediction for $\langle \sigma_x \sigma_x \sigma_x \rangle$.

Hints for Selected Problems

B.2(a): Let $\hat{a} = 0°$, $\hat{b} = \theta$, $\hat{c} = 2\theta$. Then $E(\hat{a},\hat{b}) = -\cos\theta$, $E(\hat{a},\hat{c}) = -\cos 2\theta$, $E(\hat{b},\hat{c}) = -\cos\theta$. The violation is $|-\cos\theta + \cos 2\theta| - (1 - \cos\theta)$. Maximize over $\theta$.

B.3(d): For the Werner state, $E_W(\hat{a},\hat{b}) = p \cdot E_{\text{singlet}}(\hat{a},\hat{b}) + (1-p) \cdot 0 = -p\cos\theta_{ab}$. So $S_W = p \cdot S_{\text{singlet}}$, and the maximum is $|S_W| = 2\sqrt{2}p$. Set this $> 2$ to find the threshold.

C.3(b): Expand $\hat{S}^2$ and use $\hat{A}_i^2 = \hat{B}_j^2 = \hat{I}$.

C.4(d): $F = \frac{1+2p}{3}$. Set $F > 2/3$ to get $p > 1/3$. But $p > 1/3$ only guarantees better than classical — the actual CHSH violation requires $p > 1/\sqrt{2}$.

C.5(c): The LHV argument: if $\hat{\sigma}_x^{(1)}\hat{\sigma}_y^{(2)}\hat{\sigma}_y^{(3)} = +1$ for all three cyclic permutations, then the product $\hat{\sigma}_x^{(1)}\hat{\sigma}_x^{(2)}\hat{\sigma}_x^{(3)} \cdot (\hat{\sigma}_y^{(1)})^2 \cdot (\hat{\sigma}_y^{(2)})^2 \cdot (\hat{\sigma}_y^{(3)})^2 = (+1)^3$, so $\hat{\sigma}_x^{(1)}\hat{\sigma}_x^{(2)}\hat{\sigma}_x^{(3)} = +1 \cdot 1 \cdot 1 \cdot 1 = +1$... but track the signs more carefully.