CHSH Correlation Geometry
Assemble a Bell correlation inequality and see how settings and white noise determine violation.
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CHSH Correlation Geometry
Concept Overview
The CHSH inequality is a statistical test used to demonstrate quantum entanglement and violation of local hidden-variable theories. In this module, we examine Werner-mixture correlations E(α, β) = -p cos(α - β), combining a singlet fraction p with a white-noise remainder.
Worked Example
Let's calculate the CHSH value for a pure singlet state (p = 1) with Alice measuring at 0^\circ, 90^\circ and Bob at 45^\circ, -45^\circ.
- For Alice at 0^\circ and Bob at 45^\circ, E(0, 45^\circ) = -1 · cos(-45^\circ) ≈ -0.7071.
- For Alice at 0^\circ and Bob at -45^\circ, E(0, -45^\circ) = -1 · cos(45^\circ) ≈ -0.7071.
- For Alice at 90^\circ and Bob at 45^\circ, E(90^\circ, 45^\circ) = -1 · cos(45^\circ) ≈ -0.7071.
- For Alice at 90^\circ and Bob at -45^\circ, E(90^\circ, -45^\circ) = -1 · cos(135^\circ) ≈ 0.7071.
- The CHSH combination S = |E(0, 45^\circ) + E(0, -45^\circ) + E(90^\circ, 45^\circ) - E(90^\circ, -45^\circ)|.
- Substituting values: S = |-0.7071 - 0.7071 - 0.7071 - 0.7071| = 2.8284.
The expected value is 2√2 ≈ 2.8284, which exceeds the local hidden variable bound of 2, confirming a violation.
Interactive Experiment
You can reproduce this behavior using the CHSH Correlation Geometry module.
- Set the Singlet fraction to exactly
1. - Set the Bob angle degrees to
45. - Observe the four individual correlation components in the output panel.
- Read the resulting CHSH absolute value, which will show roughly
2.8284, indicating a violation. - Try reducing the Singlet fraction to
0.5. Notice that the CHSH value drops below 2, and the system no longer violates the inequality.
Limitations and Assumptions
This model assumes coplanar spin settings. A selected setting failing the CHSH inequality is not proof of separability, nor does it serve as a complete Bell locality test across all possible observables. Additionally, individual random outcomes remain entirely unpredictable and there is no faster-than-light signalling.
References
For the original formulation of the CHSH inequality, see Clauser, Horne, Shimony, and Holt's paper proposing an experiment to test local hidden-variable theories: https://doi.org/10.1103/PhysRevLett.23.880.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive CHSH Correlation Geometry module.
Try CHSH Correlation Geometry on Riano →