| Clauser–Horne–Shimony–Holt | |
|---|---|
| Name | Clauser–Horne–Shimony–Holt |
| Caption | Schematic of a Bell-type test related to the CHSH inequality |
| Date | 1969 (inequality introduced) |
| Field | Quantum foundations |
| Related | John Bell, Bell test experiments, Bell's theorem |
Clauser–Horne–Shimony–Holt
Clauser–Horne–Shimony–Holt (commonly abbreviated CHSH) is a specific formulation of a Bell inequality introduced to provide an experimentally testable constraint distinguishing quantum mechanical predictions from those of local hidden variable theories. The CHSH inequality became central to empirical investigations of entanglement and local realism and underpins modern work in quantum information science and device-independent protocols. Its violations by quantum systems demonstrate nonclassical correlations that have broad philosophical and practical consequences.
The CHSH inequality was proposed in 1969 by John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt as an adaptation and experimentally accessible form of Bell's theorem (1964) originally formulated by John Bell. Building on earlier debates including the EPR paradox by Albert Einstein, Boris Podolsky, and Nathan Rosen, CHSH translated foundational questions about hidden variable theory and locality into measurable statistical bounds. The work of Clauser and colleagues was influenced by experimental advances at institutions such as University of California, Berkeley, Harvard University, and later implementations at laboratories like Bell Labs and Jet Propulsion Laboratory, which enabled photon-pair and atomic tests of quantum correlations. The CHSH form facilitated the transition from philosophical argument to laboratory experiment, shaping the trajectory of quantum foundations through the late 20th century.
The CHSH scenario considers two spatially separated parties, conventionally called Alice and Bob, who choose between two measurement settings (A0, A1 and B0, B1) with binary outcomes ±1. The CHSH combination S is defined as S = E(A0,B0) + E(A0,B1) + E(A1,B0) − E(A1,B1), where E denotes expectation values of product outcomes. Under any local hidden variable model satisfying locality and realism, the inequality |S| ≤ 2 holds. Quantum mechanics, however, predicts that certain entangled states such as the Bell state (maximally entangled two-qubit singlet) can achieve up to S = 2√2 (the Tsirelson bound), violating the classical bound. The derivation uses deterministic assignment of outcomes conditioned on a shared hidden variable λ and integrates over probability distributions p(λ), with locality enforcing independence of each party's outcome on the other's measurement choice.
Early experimental tests of CHSH-type inequalities were performed in the 1970s and 1980s, notably by Clauser and Stuart Freedman and later by Alain Aspect's group at École Polytechnique in the 1980s using polarization-entangled photons, providing significant evidence for violation of local realism. Advances in single-photon sources, entangled trapped ions (e.g., experiments at NIST), superconducting qubits (e.g., at IBM and Google research labs), and satellite-based tests (e.g., Micius (satellite)) have repeatedly observed CHSH violations approaching the Tsirelson bound. These experiments draw on technologies like spontaneous parametric down-conversion, beam splitters, and fast random number generators for measurement setting choices. The accumulation of data across optical, atomic, and solid-state platforms established CHSH violation as robust and reproducible.
Violations of the CHSH inequality show that no theory obeying both locality and outcome-determinism (realism) can reproduce the predictions of quantum mechanics for certain entangled states, strengthening the empirical case of Bell's theorem. This challenges classical intuitions about separability and has stimulated philosophical engagement from figures associated with the philosophy of physics and interpretations such as many-worlds interpretation, de Broglie–Bohm theory, and objective collapse models. CHSH results have prompted renewed attention to issues of causality, counterfactual definiteness, and the role of measurement, and have been central in debates about the nature of quantum nonlocality versus quantum contextuality (related to the Kochen–Specker theorem).
CHSH violations are harnessed as practical resources in modern quantum technologies. Device-independent quantum key distribution (DI-QKD) and device-independent randomness generation use observed CHSH correlations as certificates of secrecy and unpredictability without trusting internal device details. Protocols in quantum cryptography, quantum certified randomness produced by companies and research groups, and entanglement-based quantum networks exploit CHSH tests for security proofs. In quantum computing, CHSH benchmarks can assess entanglement generation and noise in platforms developed by organizations like Rigetti, IonQ, and research groups at MIT and Caltech.
Despite strong empirical support, CHSH experiments historically faced practical "loopholes" that critics argued could preserve local realism: the detection (or fair-sampling) loophole, the locality (or communication) loophole, and the freedom-of-choice (or measurement-independence) loophole. Closing these required high-efficiency detectors, space-like separation of measurement events, and independent randomness sources; notable "loophole-free" Bell tests in 2015 (e.g., groups at Delft University of Technology, NIST, and Vienna) addressed many concerns. Ongoing debates concern superdeterminism, retrocausality, and resource requirements for device independence, often engaging both physicists at institutions like Perimeter Institute and philosophers of science.
The CHSH inequality sits within a broader family of Bell inequalities and has been generalized to multipartite and higher-dimensional systems, leading to forms such as the Mermin inequality, CH inequality (Clauser–Horne), and CGLMP inequalities for d-level systems. Connections to convex geometry and polytope theory (the local polytope) formalize classical bounds, while semidefinite programming methods and hierarchy techniques (e.g., the NPA hierarchy) characterize quantum correlations and upper bounds like Tsirelson's bound. These mathematical tools inform both foundational studies and practical certification tasks in quantum information.
Category:Quantum mechanics Category:Bell inequalities Category:Quantum information theory