Questions: Bell Inequalities and Their Violation

Question 1

An experiment measures a CHSH value of 2.7. What does this result demonstrate?

• A. The particles must have communicated faster than light during measurement
• B. The correlations cannot be explained by any local hidden-variable theory
• C. The experiment has violated Heisenberg's uncertainty principle
• D. Quantum mechanics predicts a CHSH maximum of 2.7, so this result is expected classically

Answer: (B)

The CHSH bound for any local hidden-variable theory is ≤2. A measured value of 2.7 exceeds this bound, which rules out local realism — there is no way to assign pre-set hidden variables to the particles that would reproduce such correlations. Importantly, this does NOT imply faster-than-light communication; the correlations arise from entanglement but cannot be used to transmit information.

Question 2

What is the significance of the Tsirelson bound (2√2 ≈ 2.83) in quantum mechanics?

• A. It is the maximum CHSH value achievable by any physical system, including hypothetical post-quantum theories
• B. It is the classical upper bound that local hidden-variable theories must respect
• C. It is the maximum CHSH value that any quantum state can achieve — the ceiling of quantum correlations
• D. It is the exact correlation value that a maximally entangled state always produces regardless of measurement angles

Answer: (C)

The Tsirelson bound is the maximum CHSH value quantum mechanics permits, achieved by maximally entangled qubits at optimal measurement angles. It is not a universal limit for all conceivable physical theories — some mathematical frameworks exceed it — but no quantum state can. Option B confuses it with the local hidden-variable bound of 2, which is the lower benchmark. Option D is wrong because the Tsirelson bound is a maximum, not a fixed output: sub-optimal measurement angles yield lower values.

Question 3

Experimental violation of the CHSH inequality proves that entangled particles communicate faster than light at the moment of measurement.

• T. True
• F. False

Answer: False

This is the most common misreading of Bell test results. Violating the CHSH inequality rules out local hidden-variable explanations, but it does NOT imply faster-than-light communication. The correlations cannot be used to send any signal — Alice's outcomes appear random to her, and she learns nothing about Bob's setting until they compare notes classically. The 2015 loophole-free Bell experiments confirmed non-local correlations without enabling superluminal information transfer.

Question 4

Bell test experiments are agnostic about which interpretation of quantum mechanics is correct — they rule out local realism but do not distinguish between Copenhagen, many-worlds, or other interpretations.

• T. True
• F. False

Answer: True

Bell inequality violations establish that no local hidden-variable theory can reproduce all quantum predictions — but this constraint is compatible with multiple interpretations. Copenhagen (wave function collapse), many-worlds (branching), pilot-wave theory (non-local hidden variables), and others all reproduce the quantum predictions and therefore all violate the CHSH bound in the way experiments observe. The experiments close off local determinism, not the broader interpretation question.

Question 5

Why can a CHSH value greater than 2 not be explained by particles carrying 'pre-set instructions' from their shared source?

Model answer: If each particle carried predetermined answers to both possible measurement settings (A or A′, B or B′), a simple algebraic argument shows the CHSH combination |⟨AB⟩ + ⟨AB′⟩ + ⟨A′B⟩ − ⟨A′B′⟩| can never exceed 2, no matter what the pre-set values are. The bound follows from the constraint that each variable is ±1 and assignments are fixed at emission, independent of the other particle's measurement. A measured value above 2 therefore means the outcomes cannot have been pre-determined locally — the correlations must arise from the quantum state itself.

The CHSH derivation is essentially a constraint on products of ±1 variables. If A, A′, B, B′ are all ±1 and locally fixed, then AB + AB′ + A′B − A′B′ = A(B + B′) + A′(B − B′). Since B and B′ are ±1, either B + B′ = ±2 and B − B′ = 0, or vice versa. In either case the expression equals ±2. Averaging over many trials cannot exceed 2. Quantum correlations violate this because the outcomes are not locally pre-set — they are generated by the measurement process acting on a non-separable joint state.