| CHSH inequality | |
|---|---|
| Name | CHSH inequality |
| Discovered by | John F. Clauser, Michael A. Horne, Abner Shimony, Richard A. Holt |
| Discovered | 1969 |
| Field | Quantum mechanics, quantum information |
| Related | Bell's theorem, Bell test experiments, entanglement |
CHSH inequality
The CHSH inequality is a quantitative formulation of a Bell-type inequality for correlations among measurements on two separated systems. Introduced by John F. Clauser, Michael A. Horne, Abner Shimony and Richard A. Holt in 1969, it provides an experimentally accessible criterion distinguishing predictions of local realism from those of quantum mechanics. Violations of the CHSH inequality in laboratory tests have deepened understanding of entanglement and underpin technologies in quantum information science and quantum cryptography.
The CHSH inequality is central to the empirical study of nonlocal correlations predicted by Bell's theorem. It reformulates Bell's original argument into a form suitable for experiments with two observers (often named Alice and Bob) choosing between two binary-valued measurements. The inequality is relevant to foundational debates involving figures such as Albert Einstein, Niels Bohr, and modern interpreters like John Bell and Abner Shimony. Its significance extends to institutions and laboratories which performed landmark tests, including experiments at University of California, Berkeley, University of Innsbruck, University of Geneva, National Institute of Standards and Technology (NIST) and Delft University.
In the CHSH scenario two parties measure dichotomic observables A0, A1 (for Alice) and B0, B1 (for Bob). Defining correlators E(Ai,Bj) = ⟨AiBj⟩, the CHSH combination is S = E(A0,B0) + E(A0,B1) + E(A1,B0) - E(A1,B1). Under any local hidden variable model constrained by locality and realism, |S| ≤ 2. Quantum mechanics predicts that for certain entangled states, notably the maximally entangled Bell state or singlet state, the value can reach Tsirelson's bound |S| ≤ 2√2, first derived by Boris Tsirelson (often spelled Cirel'son). The inequality is closely related to mathematical constructs in probability theory, operator theory, and the study of convex sets such as the local polytope versus the quantum set.
Experimental tests of the CHSH inequality have progressed from early optical and atomic experiments to loophole-free demonstrations. Pioneering experiments include those by Clauser and Stuart Freedman, and later precision tests by Aspect's group at École Normale Supérieure, which first reported strong violations. Modern loophole-free CHSH tests have been performed by groups led by Anton Zeilinger (Innsbruck), Nicolas Gisin (Geneva), and teams at Delft University of Technology (Hannes Bernien, Ronald Hanson) and National Institute of Standards and Technology (NIST) using photons, ions, and superconducting circuits. These experiments address major loopholes: the detection loophole, the locality (or communication) loophole, and the freedom-of-choice (or setting-independence) loophole. Violations consistent with quantum predictions have been used to certify entanglement in protocols developed by researchers at companies and projects such as ID Quantique and the Quantum Internet Alliance.
Violations of the CHSH inequality challenge the conjunction of locality and classical realism, reshaping philosophical debates and public narratives about agency and causation in physics. From a justice-oriented perspective, the history of CHSH-related research underscores issues of equitable recognition, access to resources, and global participation in fundamental science: many contributions by marginalized scientists and institutions have been underrepresented in mainstream accounts. The apparatus of CHSH tests also calls attention to the political economy of big science—funding priorities at agencies such as the National Science Foundation and European Research Council influence who can perform definitive experiments. A socially aware historiography emphasizes collaborative and international efforts that democratize access to quantum technologies and distribute benefits of applications like quantum cryptography.
CHSH violations provide device-independent witnesses of entanglement, enabling protocols in device-independent quantum key distribution (DIQKD) and randomness generation. Certification based on CHSH statistics underpins security proofs that do not require trust in internal device workings, relevant to commercial projects and standards by firms like ID Quantique and research centers such as QuTech. Applications include certified private randomness production used in cryptographic systems, and subroutines in protocols for quantum teleportation and quantum networks. The inequality also informs complexity-theoretic results linking nonlocality to advantages in communication tasks studied in quantum communication complexity.
Beyond CHSH, many generalizations exist: multipartite Bell inequalities (e.g., Mermin inequality, Svetlichny inequality), inequalities for higher-dimensional systems such as the CGLMP inequality, and continuous-variable analogues. The CHSH framework connects to the study of nonlocal games like the CHSH game in theoretical computer science and to entropic inequalities in information theory. Research on self-testing uses CHSH and its variants to robustly certify quantum states and measurements. Mathematical extensions involve characterizing the boundary between classical, quantum, and post-quantum (e.g., PR box) correlations, with links to work at institutions like Perimeter Institute for Theoretical Physics, Institute for Quantum Computing (IQC), and authors including Sandu Popescu and Jonathan Barrett.
Category:Quantum mechanics Category:Bell inequalities Category:Quantum information theory