page_title: CHSH inequality
section_title: A typical CHSH experiment
hierarchical_section_title: CHSH inequality / Statement / A typical CHSH experiment
context_page_description: In physics, the CHSH inequality can be used in the proof of Bell's theorem, which states that certain consequences of entanglement in quantum mechanics cannot be reproduced by local hidden variable theories. Experimental verification of violation of the inequalities is seen as experimental confirmation that nature cannot be described by local hidden variables theories. CHSH stands for John Clauser, Michael Horne, Abner Shimony, and Richard Holt, who described it in a much-cited paper published in 1969. They derived the CHSH inequality, which, as with John Bell's original inequality, is a constraint on the statistics of "coincidences" in a Bell test experiment which is necessarily true if there exist underlying local hidden variables. This constraint can, on the other hand, be infringed by quantum mechanics.
context_section_description: In practice most actual experiments have used light rather than the electrons that Bell originally had in mind.  The property of interest is, in the best known experiments (Aspect, 1981-2), the polarisation direction, though other properties can be used.  The diagram shows a typical optical experiment.  Coincidences (simultaneous detections) are recorded, the results being categorised as '++', '+−', '−+' or '−−' and corresponding counts accumulated.
  Four separate subexperiments are conducted, corresponding to the four terms  in the test statistic S (2, above).  The settings a, a′, b and b′ are generally in practice chosen to be 0, 45°, 22.5° and 67.5° respectively — the "Bell test angles" — these being the ones for which the QM formula gives the greatest violation of the inequality.
  For each selected value of a and b, the numbers of coincidences in each category  are recorded.  The experimental estimate for  is then calculated as:
  Once all the 's have been estimated, an experimental estimate of S (2) can be found.  If it is numerically greater than 2 it has infringed the CHSH inequality and the experiment is declared to have supported the QM (Quantum Mechanics) prediction and ruled out all local hidden variable theories.
  The CHSH paper lists many preconditions (or "reasonable and/or presumable assumptions") to derive the simplified theorem and formula. For example, for the method to be valid, it has to be assumed that the detected pairs are a fair sample of those emitted. In actual experiments, detectors are never 100% efficient, so that only a sample of the emitted pairs are detected. A subtle, related requirement is that the hidden variables do not influence or determine detection probability in a way that would lead to different samples at each arm of the experiment.
caption: Schematic of a "two-channel" Bell test
  The source S produces pairs of photons, sent in opposite directions. Each photon encounters a two-channel polariser whose orientation can be set by the experimenter. Emerging signals from each channel are detected and coincidences counted by the coincidence monitor CM.