| Clauser–Horne (CH) inequality | |
|---|---|
| Name | Clauser–Horne inequality |
| Field | Quantum physics |
| Introduced | 1974 |
| Introduced by | John F. Clauser and Michael A. Horne |
| Related | Bell's theorem; Bell inequalities; local hidden-variable theories |
Clauser–Horne (CH) inequality
The Clauser–Horne (CH) inequality is a quantitative constraint on correlations predicted by any local realistic theory for measurements on spatially separated systems. Introduced by John F. Clauser and Michael A. Horne in 1974, it refines earlier Bell's theorem formulations to account for detector inefficiencies and provides an experimentally accessible inequality used to test quantum predictions against local realism. Violations of the CH inequality by quantum experiments have deep consequences for our understanding of entanglement and nonlocality.
The CH inequality occupies a central role in the study of quantum nonlocality and the empirical testing of foundational questions raised by Albert Einstein, Boris Podolsky, and Nathan Rosen in the EPR paradox. It is closely related to other Bell inequality variants such as the CHSH inequality and the original Bell inequalities but is tailored to scenarios with incomplete detection and asymmetric settings. The inequality links to experiments performed by groups led by Clauser, Stuart Freedman, Alain Aspect, Anton Zeilinger, and later groups at institutions like University of California, Berkeley, University of Innsbruck, and corporate and national labs that implemented loophole-free tests.
The CH inequality is derived within the framework of local hidden-variable theorys by assuming measurement outcomes are determined by shared hidden variables λ with a probability distribution ρ(λ). For two observers, typically labeled A and B, with binary detection events at settings a, a' and b, b', the CH inequality constrains joint and marginal probabilities P(A, B|a, b), P(A|a), P(B|b). One common form is: P(A,B|a,b) + P(A,B|a,b') + P(A,B|a',b) − P(A,B|a',b') − P(A|a') − P(B|b) ≤ 0. This inequality is testable without fair-sampling assumptions because it uses directly measured single and coincidence count rates. The quantum mechanical prediction for entangled states such as the singlet state can violate this bound, with the degree of violation computed using quantum states, projection measurements, and the Born rule.
The CH inequality rests on explicit assumptions: (1) locality — that measurement outcomes at one wing do not causally influence the other faster than light; (2) realism — that outcomes are determined by pre-existing properties encoded in λ; and (3) measurement independence (sometimes called freedom of choice) — that settings are uncorrelated with λ. A major motivation for the CH formulation was to relax or avoid the fair-sampling assumption used in earlier optical tests, making the inequality robust against detector inefficiencies. Discussions of dropouts, detection loophole, and setting dependence link the CH approach to debates over closing the major experimental loopholes that permit local realistic explanations.
Experimental tests of the CH inequality have been performed using entangled photons, trapped ions, atoms, and solid-state systems. Early optical tests by John Clauser and collaborators used coincidence-counting electronics; later, the experiments of Alain Aspect introduced time-varying analyzers to address locality. More recent loophole-free Bell tests that closed detection and locality loopholes — such as those by groups including Ronald Hanson at Delft University of Technology and others at NIST and IQOQI Innsbruck — used CH-type or CHSH-type criteria to demonstrate violations consistent with quantum mechanics. Experimental implementations must control for background noise, accidental coincidences, and signaling; statistical analysis often employs hypothesis testing tailored to the CH framework.
Violations of the CH inequality rule out broad classes of local hidden-variable models that satisfy its assumptions. This reinforces the central conclusions of Bell's theorem: no theory that maintains both locality and certain forms of realism can reproduce all quantum predictions. Alternative models — including nonlocal hidden-variable theories like de Broglie–Bohm theory and superdeterministic proposals — evade the CH constraints by rejecting one or more assumptions (e.g., locality or measurement independence). The CH results thus sharpen philosophical debates about causation, counterfactual definiteness, and the ontology of the quantum state in foundations research.
The CH inequality is directly connected to the operational resources of entanglement and to measures of quantum nonlocality used in quantum information theory. Violations are employable as device-independent witnesses of entanglement and security in protocols such as quantum key distribution (QKD). CH-style inequalities inform device-independent certification, randomness generation, and quantum cryptographic proofs by providing experimentally testable constraints that do not rely on detailed trust in measuring devices. Links to concepts like Bell nonlocality, entanglement swapping, and resource theories of nonlocality make the CH inequality relevant across applied and theoretical branches of quantum technology.
Foundational experiments testing the CH inequality have social and philosophical resonance beyond physics: they challenge classical intuitions about separability, causation, and individual autonomy of systems. The experimental enterprise has demanded equitable access to instrumentation and collaboration across universities, national labs, and industry — raising questions about research funding, scientific inclusion, and the global distribution of technological benefits. Philosophically, CH-related violations prompt reflection on scientific realism, the role of experiment in adjudicating metaphysical claims, and ethical considerations in deploying quantum technologies (e.g., surveillance or cryptography) that rely on entanglement. Advocates for responsible innovation emphasize that foundational advances should be paired with policies ensuring that quantum technologies advance justice, reduce inequality, and respect civil liberties.
Category:Quantum mechanics Category:Bell inequalities Category:Quantum information science