CHSH Inequality and Bell Tests

qc/entanglement qc/math

The experimental proof that entanglement isn’t just “correlation from a hidden pre-agreed answer.” Any theory where each particle secretly carries a definite, pre-determined value (a local hidden variable) is bound by the CHSH inequality: $S\le 2$, for a specific combination $S$ of correlation measurements Alice and Bob make with independently-chosen settings. Quantum mechanics predicts — and experiments confirm — $S\approx 2\sqrt{2}\approx 2.83$, clearly violating the classical bound.

CHSH = Clauser, Horne, Shimony, Holt. First experimentally confirmed by Alain Aspect (1982); loophole-free versions closed remaining experimental gaps in 2015.

Key insight: “Entanglement is not ignorance about a pre-existing state — the CHSH violation is the experimental receipt.” Classically, correlated outcomes always come from either direct causation or a shared hidden fact fixed in advance. The CHSH violation rules out the second option experimentally, not just theoretically.

Why this doesn’t allow signaling

Measuring one particle instantly affects the joint statistics, but this cannot send information — the correlation is only visible once Alice and Bob classically compare their individual (locally random) results, and that comparison is itself light-speed-limited. The non-locality lives in the correlations between two already-random outcomes, not in either individual outcome — so nothing travels faster than light. This is the same resolution as Quantum Teleportation’s reliance on a classical channel.

Related

• Bell States — the states used to violate CHSH
• Quantum Teleportation
• No-Cloning Theorem
• Measurement and Collapse
• E91 — Entanglement-Based Quantum Key Distribution — this exact $S\le 2$/$S\approx 2\sqrt{2}$ math, reused as a security primitive

Self-Check

• What does the classical CHSH bound ($S\le 2$) assume about how the world works, and what does violating it rule out?
• Could you explain why entanglement correlations don’t allow faster-than-light signaling, even though measuring one particle instantly affects the joint state?
• Why did it take until 1982 (Aspect) and 2015 (loophole-free) to fully confirm something quantum mechanics predicted decades earlier?
• If you parameterize a Bell state with a single rotation angle $\theta$ (e.g. Ry(θ) on one qubit before the entangling gate) and sweep $\theta$ while computing $S$, why does that let you scan continuously between CHSH-violating and non-violating configurations, rather than jumping straight from “classical” to “maximally quantum”?
• Framed as a nonlocal game (Alice and Bob win if $a\oplus b=x\wedge y$ for their random inputs $x,y$), why does the best classical strategy cap out at 75% win probability while the best quantum strategy reaches ${\cos }^2(\pi /8)\approx 85.4\%$ — and how does that number relate to $S\approx 2\sqrt{2}$?