| Clauser–Horne–Shimony–Holt (CHSH) inequality | |
|---|---|
| Name | CHSH inequality |
| Field | Quantum mechanics |
| Introduced | 1969 |
| Authors | John F. Clauser; Michael A. Horne; Abner Shimony; Richard A. Holt |
| Related | Bell's theorem, EPR paradox, Tsirelson bound |
Clauser–Horne–Shimony–Holt (CHSH) inequality
The Clauser–Horne–Shimony–Holt (CHSH) inequality is a specific testable form of Bell's theorem that provides constraints on correlations predicted by any theory obeying local realism. It translates philosophical debates about the EPR paradox and hidden variables into experimentally accessible statistical inequalities, and thus underpins empirical demonstrations of quantum nonlocality with profound consequences for quantum information and the politics of scientific practice.
The CHSH inequality was published in 1969 by John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt as an adaptation of earlier work by John Stewart Bell (Bell's inequality, 1964). It arose amid renewed interest in the Einstein–Podolsky–Rosen paradox (EPR) and stochastic hidden-variable models that attempted to preserve locality and realism. The formulation offered clear experimental prescriptions that motivated pioneering optical tests by Stuart Freedman and John Clauser and later refined experiments by Alain Aspect, and modern loophole-free tests at institutions such as Delft University of Technology, University of Vienna and National Institute of Standards and Technology (NIST). The CHSH form remains the canonical bridge between theoretical constraints and laboratory verification.
The CHSH scenario involves two spatially separated observers, traditionally named Alice and Bob, who each choose one of two measurement settings (A0, A1 and B0, B1) with binary outcomes ±1. Define the CHSH combination: S = E(A0,B0) + E(A0,B1) + E(A1,B0) - E(A1,B1), where E(Ai,Bj) denotes the expectation value of the product of outcomes for settings i and j. Under any local hidden-variable theory satisfying locality and outcome determinism, the inequality |S| ≤ 2 must hold. Quantum mechanics, using entangled states such as the singlet state of two spin-1/2 particles or polarization-entangled photons, predicts correlations that can yield |S| up to 2√2, the Tsirelson bound.
The CHSH inequality is derived by assuming a common hidden variable λ distributed with probability ρ(λ), and deterministic outcome functions A(a,λ), B(b,λ) ∈ {±1} for measurement settings a,b. Linearity and algebraic manipulation of products A(a,λ)B(b,λ) produce the CHSH bound after integrating over λ. The result is a direct corollary of Bell's theorem: no local realistic theory can reproduce all quantum predictions. The CHSH form emphasizes operational assumptions—choice independence, parameter independence, and outcome independence—and clarifies which empirical facts force rejection of local realism. The inequality is often presented alongside alternative formulations such as the Clauser–Horne inequality and the original Bell inequality to highlight differing assumptions about fair sampling and detector efficiencies.
Quantum theory violates the CHSH bound for appropriate entangled states and measurement bases. For the two-qubit singlet state and measurements in mutually rotated bases, the quantum expectation yields S = 2√2, the maximum allowed by quantum theory, known as the Tsirelson bound derived by Boris Tsirelson (Cirel'son). Values of S between 2 and 2√2 demonstrate quantum nonlocality but still respect no-signalling constraints encoded in the popescu–rohrlich box thought experiments. Studies connecting CHSH violations to entanglement measures (e.g., concurrence, negativity) and to device-independent protocols establish CHSH as a quantitative resource in quantum theory.
CHSH-style experiments have been performed using entangled photons, trapped ions, superconducting qubits, and atomic ensembles. Landmark experiments addressing locality and detection loopholes include those by Alain Aspect (1982), Weihs et al. (1998), and the 2015 loophole-free tests by teams at Delft University of Technology, NIST, and the University of Vienna with partner institutions. Major experimental concerns include the detection (efficiency) loophole, the locality (communication) loophole, and the freedom-of-choice (measurement independence) loophole. Addressing these requires fast random-number generators, space-like separation of measurement stations, and high-efficiency detectors—technical demands that reflect resource disparities among laboratories and nations.
Violations of the CHSH inequality have practical and theoretical implications across quantum cryptography, quantum teleportation, and device-independent quantum information. Device-independent quantum key distribution and randomness expansion protocols rely on observed S > 2 to certify security without trusting internal device details. Foundationally, CHSH results motivate exploration of alternative ontologies (e.g., de Broglie–Bohm theory, objective collapse models) and inform reconstructions of quantum theory from information-theoretic axioms. The inequality thus plays a central role in translating foundational insight into technology.
The empirical refutation of local realism via CHSH tests raises philosophical questions about causation, agency, and scientific authority. Nonlocal correlations have been invoked in debates over scientific realism and the limits of mechanistic explanation. Ethically, the distribution of capabilities to perform cutting-edge quantum experiments reflects inequities in funding, infrastructure, and access to talent; democratizing participation in foundational science intersects with broader social justice concerns. The deployment of CHSH-enabled technologies—secure communication, quantum random number generators, and novel sensors—also prompts policy discussions about surveillance, equity, and responsible innovation led by universities, national labs, and industry actors.
Category:Foundations of quantum mechanics Category:Quantum information theory