Clauser–Horne–Shimony–Holt

Clauser–Horne–Shimony–Holt is a formulation of a Bell-type inequality used to test correlations predicted by Quantum mechanics against those allowed by Local realism in experiments with entangled particles. It was proposed by John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt in 1969 and has been central to experimental and theoretical work involving John Stewart Bell, Alain Aspect, Anton Zeilinger, Nicolas Gisin, and Sergio F. Barbosa. The formulation provides a practical inequality for two-party, two-setting, two-outcome scenarios and has influenced tests at laboratories such as Bell Labs, Los Alamos National Laboratory, University of Innsbruck, Harvard University, and University of Vienna.

Background and theoretical formulation

The inequality arises in the context of debates involving John Bell and his 1964 theorem, debates later taken up by figures including David Bohm, Albert Einstein, Boris Podolsky, Nathan Rosen, and critics like Einstein–Podolsky–Rosen paradox. Clauser and colleagues sought a testable form of Bell's ideas suitable for experiments performed by teams at institutions such as Stanford University, Columbia University, Massachusetts Institute of Technology, and California Institute of Technology. The CHSH scenario assumes two observers often named Alice and Bob, drawing on nomenclature popularized in works by Charles H. Bennett and Gilles Brassard in quantum information theory contexts such as BB84 protocol. The formulation uses correlations between measurement outcomes tied to settings labeled a, a', b, b' and relates to earlier theoretical work by Frans Kramers, Hendrik Lorentz, and the later formal apparatus codified by Paul Dirac and Werner Heisenberg.

CHSH inequality and mathematical derivation

The CHSH inequality is typically written in terms of expectation values E(a,b) for measurement settings a and b and yields a bound of 2 for any theory respecting the assumptions of local causality and predetermined outcomes, assumptions debated in writings by Hans Reichenbach, Karl Popper, and Imre Lakatos. Quantum predictions for a maximally entangled singlet state, discussed by John Clauser and observed in frameworks advanced by John Wheeler and Richard Feynman, can attain values up to 2√2, often referred to as the Tsirelson bound associated with Boris Tsirelson. The derivation employs algebraic relations used in foundational analyses by Eugene Wigner, Arthur Wightman, and Julian Schwinger, and is linked mathematically to studies by John von Neumann and Norbert Wiener on operator bounds.

Experimental tests and implementations

Early experiments testing the CHSH inequality were performed by teams including John Clauser and Stuart Freedman at University of California, Berkeley, and later decisive tests were conducted in campaigns led by Alain Aspect at École Normale Supérieure and by groups at Weizmann Institute of Science, University of Geneva, Trinity College Dublin, and University of Maryland. Modern implementations exploit platforms from Aspect's experiments to photon-pair sources using spontaneous parametric down-conversion, technologies developed at Bell Labs and refined in laboratories such as MIT Lincoln Laboratory, Max Planck Institute for Quantum Optics, NIST, and Quantum Optics Group at University of Vienna. Recent loophole-free demonstrations involved collaborations among teams at Delft University of Technology, NIST, University of Vienna, University of Waterloo, and institutions affiliated with Microsoft and Google, combining entanglement distribution methods from satellite experiments like those by Micius (satellite) and fiber networks demonstrated by Telecom companies.

Implications for local realism and quantum mechanics

Violations of the CHSH inequality played a decisive role in undermining classical realist positions advocated historically by Albert Einstein and allies, and supported quantum nonlocality narratives advanced by researchers including Niels Bohr, Max Born, Paul Dirac, and modern proponents like Bell. The empirical results informed interpretations championed by Copenhagen interpretation figures and motivated alternative frameworks proposed by David Bohm (pilot-wave theory), Hugh Everett (many-worlds interpretation), and objective-collapse models advanced by Ghirardi–Rimini–Weber and Penrose. The experimental record also influenced practical developments in quantum cryptography and device-independent protocols by researchers such as Antonio Acín, Valerio Scarani, and Renato Renner, and has been applied to tests of causal structure studied by Judea Pearl and Lucien Hardy.

Variants, extensions, and related inequalities

Extensions of the CHSH framework include multipartite inequalities like those by Daniel Greenberger, Michael Horne, and Anton Zeilinger (GHZ), as well as multipartite formulations by Svetlichny, Mermin, and Ardehali. Related two-party inequalities encompass those by John Bell, Clauser–Horne, Eberhard, and Leggett, while generalizations to high-dimensional systems were developed by Durt, Kaslikowski, and Collins (CGLMP). Device-independent certification, randomness expansion, and self-testing approaches leveraging CHSH violations have been advanced by groups around Umesh Vazirani, Stefano Pironio, Jean-Daniel Bancal, and Antonio Acín. Mathematical links exist between CHSH-type bounds and concepts in operator algebras studied by Alain Connes and combinatorial frameworks explored by Lovász and Paul Erdős.

Category:Quantum mechanics