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CHSH Inequality (S-Parameter) Calculator

Measure used to prove quantum entanglement.

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S-Parameter

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Overview

The CHSH inequality is a mathematical formulation used in quantum mechanics to test Bell's theorem, providing a criterion to distinguish between local hidden variable theories and quantum entanglement. It calculates the S-parameter based on correlation values of particle properties measured at different detector orientations.

Symbols

Variables

S = S-Parameter, E_1 = E(a,b), E_2 = E(a,b'), E_3 = E(a',b), E_4 = E(a',b')

S-Parameter
E(a,b)
E(a,b')
E(a',b)
E(a',b')

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When To Use

When to use: Use this formula when conducting Bell tests to verify the presence of non-local correlations in a bipartite system. It assumes that two observers, typically named Alice and Bob, each choose between two measurement settings for their respective particles.

Why it matters: It serves as the experimental proof that the universe does not obey local realism, as quantum systems can violate the classical bound of 2. This principle is fundamental for secure quantum communication and the development of quantum computers.

One free problem

Practice Problem

A quantum optics experiment measures four correlation values: E(a, b) = 0.707, E(a, b') = -0.707, E(a', b) = 0.707, and E(a', b') = 0.707. Calculate the resulting S-parameter to determine if it reaches Tsirelson's bound.

E(a,b)0.707
E(a,b')-0.707
E(a',b)0.707
E(a',b')0.707

Solve for:

Hint: Plug the values into the formula S = e1 - e2 + e3 + e4 and watch the double negative.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: CHSH inequality
  2. Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information (10th Anniversary Edition). Cambridge University Press.
  3. Clauser, J. F., Horne, M. A., Shimony, A., & Holt, R. A. (1969). Proposed Experiment to Test Local Hidden-Variable Theories.
  4. Griffiths, David J. Introduction to Quantum Mechanics. 3rd ed. Cambridge University Press, 2018.
  5. Nielsen and Chuang Quantum Computation and Quantum Information
  6. Clauser et al. Proposed Experiment to Test Local Hidden-Variable Theories
  7. Bell On the Einstein Podolsky Rosen Paradox
  8. CHSH inequality Wikipedia