CHSH Inequality

CHSH is the standard Bell inequality for two parties, two settings per party, and outcomes normalized to $\pm1$. The canonical CHSH Inequality page owns the stochastic local proof, Tsirelson-bound derivation, singlet realization, and PR-box calculation.

Helpful background. For the local, quantum, and no-signaling derivations behind this lookup card, consult the canonical CHSH Inequality page.

Definition and bounds

For correlators

$$E_{xy} = \sum_{a,b=\pm1}ab\,p(a,b\mid x,y),$$

fix the convention

$$S=E_{00}+E_{01}+E_{10}-E_{11}.$$

The three benchmark ceilings are

$$\begin{array}{c|c} \text{model class}&\text{bound}\cr \hline \text{Bell-local}&\lvert S\rvert\le2\cr \text{quantum}&\lvert S\rvert\le2\sqrt2\cr \text{arbitrary correlations}&\lvert S\rvert\le4. \end{array}$$

A Popescu–Rohrlich box is no-signaling and reaches $4$, so the algebraic ceiling is not a quantum value.

Assumptions

• The scenario has settings $x,y\in\{0,1\}$ and outcomes $a,b\in\{-1,+1\}$.
• Bell-local responses factorize conditional on a shared variable.
• The shared-variable distribution is independent of the settings.
• One sign convention for $S$ is retained throughout; local relabelings can change signs without changing the physical bound.

Interpretation

A value above $2$ rules out the declared Bell-local model for the measured distribution. It does not imply signaling. The value $2\sqrt2$ is the maximum quantum magnitude, not the value of every entangled state or every measurement choice. Physical spin outcomes $\pm\hbar/2$ must be rescaled to the dimensionless variables $\pm1$ used here.

Quick check

A model gives $S=2.4$. What is established from that number alone?

Solution

It exceeds the Bell-local bound and is compatible with the quantum bound. To interpret it as an experimental Bell violation, one still needs the declared scenario, valid estimation of the four correlators, and the physical and statistical assumptions of the test. The number alone does not establish signaling or identify a unique quantum state.

References

• J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Physical Review Letters 23, 880–884 (1969).
• B. S. Cirel’son, “Quantum generalizations of Bell’s inequality,” Letters in Mathematical Physics 4, 93–100 (1980).
• A. Fine, “Hidden variables, joint probability, and the Bell inequalities,” Physical Review Letters 48, 291–295 (1982).