| CH inequality | |
|---|---|
| Name | Clauser–Horne inequality |
| Field | Quantum mechanics |
| Formulated | 1974 |
| Authors | John F. Clauser and Michael A. Horne |
| Related | Bell's theorem |
CH inequality
The CH inequality is a constraint on measurable correlations in experiments on entangled systems derived by John F. Clauser and Michael A. Horne in 1974. It provides an experimentally testable bound that distinguishes between predictions of quantum mechanics and those of local realistic theories, playing a central role in debates over locality and hidden variable models. The inequality matters because it framed practical protocols for loophole-aware tests of quantum nonlocality and influenced foundational work in quantum information and experimental optics.
The CH inequality arose in the context of efforts to operationalize and test Bell's theorem after the seminal work of John S. Bell in 1964. Early experiments by Stuart J. Freedman and John Clauser (1972) and later by Alain Aspect brought the question into laboratory practice. Clauser and Horne published a formulation that addressed detection inefficiencies and realistic experimental constraints, building on assumptions used in the Einstein–Podolsky–Rosen paradox (EPR) debate initiated by Albert Einstein, Boris Podolsky, and Nathan Rosen. The CH inequality influenced subsequent experimental programs at institutes such as University of California, Berkeley, Bell Laboratories, CERN, and National Institute of Standards and Technology (NIST).
The Clauser–Horne inequality expresses bounds on joint and marginal detection probabilities for two separated measurement stations, often labelled A and B, with settings a, a' and b, b'. It uses observable probabilities P(A,B|a,b) and single detection probabilities P(A|a), P(B|b) rather than idealized expectation values. The canonical CH form can be written in terms of measured rates and reads, for appropriate labeling, as a linear combination of four joint probabilities and two single probabilities that must be non‑negative under any local hidden variable model satisfying realism and locality. The inequality is tailored to experiments employing photons, ion traps, or atomic ensembles where detector inefficiencies and background counts are significant. The CH formulation complements related inequalities such as the CHSH inequality (Clauser–Horne–Shimony–Holt) and the original Bell inequalities.
Researchers have used the CH inequality in experiments designed to close specific loopholes. Notable experimental efforts employing CH-style tests include those by Alain Aspect (1982), Anton Zeilinger's group (1998 onward), and more recent loophole-free tests by teams led by Saul Perlmutter—note: Perlmutter is known for other work—while teams at Delft University of Technology, Weizmann Institute of Science, NIST, and University of Vienna reported Bell tests that integrated high-efficiency detectors and space-like separation. CH-type analyses address the detection loophole (also called fair-sampling) and the locality loophole by relying directly on measured detection probabilities. Violations observed in optical experiments using parametric down-conversion sources, superconducting qubits, and trapped ions strengthen the case for quantum nonlocal correlations predicted by the singlet state and other entangled states, with practical implications for quantum cryptography protocols such as device-independent quantum key distribution.
The CH inequality is a manifestation of constraints implied by local hidden variable theories, similar in spirit to Bell's theorem but adapted for realistic detector response. Violation of the CH inequality, like violation of the CHSH inequality, implies that no theory based on both locality and predetermined outcomes (local realism) can reproduce the quantum predictions for certain entangled states. The result has philosophical and practical consequences for interpretations of quantum theory, touching on Copenhagen interpretation, de Broglie–Bohm theory, many-worlds interpretation, and objective-collapse proposals. It also informs debates in philosophy of science about realism and operational definitions of measurement.
The derivation begins with a local hidden variable model in which joint probabilities factor through a hidden variable λ distributed with density ρ(λ) ≥ 0. One assumes outcome probabilities at each wing depend only on the local setting and λ, and that joint probabilities respect normalization and non-negativity. The CH inequality follows from algebraic manipulation of these assumptions to obtain a bound: a linear inequality linking P(A,B|a,b), P(A,B|a,b'), P(A,B|a',b), P(A,B|a',b') and marginals P(A|a'), P(B|b). Crucially, the CH derivation relaxes assumptions about perfect detection and avoids requiring fair-sampling, replacing expectation-value expressions with directly observable count rates. The key assumptions to violate for quantum predictions are locality, outcome independence, or measurement realism; the mathematical structure highlights where quantum correlations depart from classical convex cones of local probability distributions.
The CH inequality remains an essential tool in foundational experiments and in the certification of nonclassical resources in quantum information science. It is used in analyses for device characterization, entanglement witnesses, and protocols requiring minimal assumptions, such as device-independent certification of randomness. The inequality influenced standards and best practices for Bell tests, informing detector technology advances at laboratories like MIT Lincoln Laboratory and Max Planck Institute for Quantum Optics. CH-based reasoning also underpins theoretical work on nonlocal games, bounds in quantum communication complexity, and the study of causal models in quantum theory. Overall, the Clauser–Horne inequality helped transition debates from philosophical abstraction to rigorous, state-of-the-art experiments that underpin modern quantum technologies and national efforts in secure communications and metrology.
Category:Quantum mechanics Category:Foundations of quantum mechanics Category:Bell inequalities