This article was accepted into the corpus but its outbound wikilinks were never NER-processed — typical at the deepest BFS hop or when the run's entity cap was reached. No expansion funnel to show.
| CHSH inequality | |
|---|---|
| Name | CHSH inequality |
| Field | Quantum physics |
| Discovered | 1969 |
| Discoverers | John Clauser; Michael Horne; Abner Shimony; Richard Holt |
| Related | Bell's theorem; Einstein–Podolsky–Rosen paradox; quantum entanglement |
CHSH inequality The CHSH inequality is a quantitative constraint on correlations predicted by any local hidden variable theory, formulated by John Clauser, Michael Horne, Abner Shimony, and Richard Holt. It refines Bell's theorem and provides an experimentally testable bound distinguishing classical realistic models from quantum mechanical predictions, notably in contexts related to the Einstein–Podolsky–Rosen paradox and laboratory tests using entangled particles such as in experiments inspired by Alain Aspect and Anton Zeilinger.
The CHSH inequality situates within the historical debate begun by Albert Einstein, Boris Podolsky, and Nathan Rosen over completeness of Quantum mechanics and was influenced by the later formalism of John Stewart Bell. The inequality established by Clauser, Horne, Shimony, and Holt provided an operational criterion applied in experiments at institutions like University of California, Berkeley, Harvard University, and Austrian Academy of Sciences. It underpins modern developments in protocols from Quantum key distribution to device-independent certification studied by research groups at Massachusetts Institute of Technology and University of Geneva.
Consider two spatially separated parties commonly named Alice and Bob operating measurement devices analogous to setups in laboratories at Bell Labs or CERN. Alice chooses between two dichotomic observables A0 and A1, while Bob chooses between B0 and B1, each yielding outcomes ±1; the CHSH correlator S is formed from expectation values E(Ai,Bj) as S = E(A0,B0) + E(A0,B1) + E(A1,B0) − E(A1,B1). Local hidden variable models constrained by premises similar to those discussed by David Bohm and Louis de Broglie impose the bound |S| ≤ 2, whereas quantum mechanics permits |S| up to 2√2 for certain states and measurement settings associated with optimal choices studied by Asher Peres and Niels Bohr.
The usual derivation employs a hidden variable λ distributed with probability ρ(λ), paralleling probabilistic approaches used in works at Princeton University and University of Chicago. One writes deterministic outcomes A_i(λ), B_j(λ) ∈ {−1,+1} and constructs the combination A0(λ)[B0(λ)+B1(λ)] + A1(λ)[B0(λ)−B1(λ)], whose absolute value is ≤ 2 pointwise. Integrating over λ with ρ(λ) yields the CHSH bound |S| ≤ 2. This argument echoes assumptions critiqued in discussions by John Bell and later clarified in analyses at Stanford University and Imperial College London concerning locality and realism.
Quantum theory, through entangled states such as the singlet state used in proposals by David Bohm and demonstrations by Alain Aspect, can violate the CHSH bound. For the maximally entangled two-qubit state |ψ−⟩ and suitable measurement axes related to rotations studied by Élie Cartan and Vladimir Fock, quantum predictions yield S = 2√2, known as the Tsirelson bound derived by Boris Tsirelson. This violation has been computed in formalisms employed in studies at Perimeter Institute and Max Planck Institute for Quantum Optics, and underlies nonlocality features discussed in reviews associated with John Preskill and Scott Aaronson.
Early pivotal tests were performed by Clauser and collaborators and later high-visibility implementations by Alain Aspect in the 1980s; subsequent loophole-closing experiments involved groups at University of Vienna and collaborations with Institute for Quantum Optics and Quantum Information. Key experimental issues include the locality loophole addressed in optical fiber and satellite experiments related to projects at European Space Agency and the detection loophole mitigated in ion-trap experiments at National Institute of Standards and Technology and superconducting platforms explored at IBM Research. Recent ‘‘loophole-free’’ tests were reported by consortia including researchers from Delft University of Technology and National University of Singapore.
Violations of the CHSH inequality have direct consequences for foundational questions considered by Albert Einstein and operational tasks in quantum technologies. They certify entanglement critical for protocols like Quantum key distribution implemented by companies collaborating with Telefónica and public projects at European Organization for Nuclear Research. Device-independent randomness generation and device-independent cryptography leverage CHSH tests in theoretical frameworks advanced at University of Copenhagen and École Normale Supérieure. Philosophical implications have been debated in venues such as Princeton University Press and Cambridge University Press concerning realism and causality.
Generalizations include multipartite Bell inequalities like the Mermin and Svetlichny inequalities explored at Los Alamos National Laboratory and multicopy or high-dimensional variants studied by teams at University of Oxford and University of Geneva. The CHSH scenario connects to resource theories of nonlocality researched at Centre national de la recherche scientifique and to entropic and steering inequalities analyzed by groups at Australian National University and University of Toronto. Mathematical extensions consider Tsirelson-type bounds and semidefinite programming techniques developed in collaborations involving École Polytechnique and IBM Research.