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L5-08 · Bell correlations and the CHSH test

Calculate a CHSH value and state its assumptions.

Level 5 icon Level 5 · Advanced undergraduate
Objective. Calculate a CHSH value and state its assumptions.
Before this lesson. L5-04, L4-02
Hands-on lab. Entanglement and Bell tests

Learn

A Bell experiment compares correlations under several local measurement choices. In the CHSH test, each laboratory chooses between two settings and records a sign. A local hidden-variable model obeys an absolute CHSH bound of two under its measurement-independence and locality assumptions.

Quantum theory permits values up to two times the square root of two. This exceeds the local bound but still respects no signalling. Neither outcome lets one laboratory choose the other outcome. A violation excludes a stated class of models, rather than every possible causal explanation.

Real tests must address detection, setting selection, timing, and statistical uncertainty. Selecting only convenient detected events can invalidate the comparison. Finite samples require a defined test statistic and uncertainty method. A simulated expected value is not an experimental loophole-free Bell test.

Practice

Solution

The signed sum is 0.6 + 0.6 + 0.6 - (-0.6) = 2.4. This exceeds two as a point value. Sample counts and a statistical test are still needed.

Check your understanding

Does a CHSH violation permit a controllable faster-than-light message?

Show the answer

C. No, quantum marginals can obey no signalling

The violation concerns local hidden-variable explanations. Quantum joint correlations can violate that bound while each local marginal stays independent of the remote setting.

Sources

This lesson draws on Bell, 1964Clauser et al., 1969Preskill, n.d..

Next: L5-09 · Entropy and quantum tomography

References
  1. Bell, J. S. (1964). On the Einstein Podolsky Rosen paradox. Physics Physique Fizika, 1(3), 195–200. 10.1103/physicsphysiquefizika.1.195
  2. Clauser, J. F., Horne, M. A., Shimony, A., & Holt, R. A. (1969). Proposed Experiment to Test Local Hidden-Variable Theories. Physical Review Letters, 23(15), 880–884. 10.1103/physrevlett.23.880
  3. Preskill, J. (n.d.). Course Information for Physics 219/Computer Science 219 Quantum Computation (Formerly Physics 229). California Institute of Technology course page with lecture notes. https://www.preskill.caltech.edu/ph219/