page_title: Bell's theorem
section_title: Testing by practical experiments
hierarchical_section_title: Bell's theorem / Testing by practical experiments
context_page_description: Bell's theorem proves that quantum physics is incompatible with local hidden variable theories. It was introduced by physicist John Stewart Bell in a 1964 paper titled "On the Einstein Podolsky Rosen Paradox", referring to a 1935 thought experiment that Albert Einstein, Boris Podolsky and Nathan Rosen used to argue that quantum physics is an "incomplete" theory. By 1935, it was already recognized that the predictions of quantum physics are probabilistic. Einstein, Podolsky and Rosen presented a scenario that, in their view, indicated that quantum particles, like electrons and photons, must carry physical properties or attributes not included in quantum theory, and the uncertainties in quantum theory's predictions are due to ignorance of these properties, later termed "hidden variables". Their scenario involves a pair of widely separated physical objects, prepared in such a way that the quantum state of the pair is entangled.
  Bell carried the analysis of quantum entanglement much further.
context_section_description: Experimental tests can determine whether the Bell inequalities required by local realism hold up to the empirical evidence.
  Actually, most experiments have been performed using polarization of photons rather than spin of electrons (or other spin-half particles). The quantum state of the pair of entangled photons is not the singlet state, and the correspondence between angles and outcomes is different from that in the spin-half set-up. The polarization of a photon is measured in a pair of perpendicular directions. Relative to a given orientation, polarization is either vertical (denoted by V or by +) or horizontal (denoted by H or by -). The photon pairs are generated in the quantum state
  where  and  denotes the state of a single vertically or horizontally polarized photon, respectively (relative to a fixed and common reference direction for both particles).
  When the polarization of both photons is measured in the same direction, both give the same outcome: perfect correlation. When measured at directions making an angle 45° with one another, the outcomes are completely random (uncorrelated). Measuring at directions at 90° to one another, the two are perfectly anti-correlated. In general, when the polarizers are at an angle θ to one another, the correlation is cos(2θ). So relative to the correlation function for the singlet state of spin half particles, we have a positive rather than a negative cosine function, and angles are halved: the correlation is periodic with period π instead of 2π.
  Bell's inequalities are tested by "coincidence counts" from a Bell test experiment such as the optical one shown in the diagram.  Pairs of particles are emitted as a result of a quantum process, analysed with respect to some key property such as polarisation direction, then detected.  The setting (orientations) of the analysers are selected by the experimenter.
  Bell test experiments to date overwhelmingly violate Bell's inequality.
caption: Scheme 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 (a or b) can be set by the experimenter. Emerging signals from each channel are detected and coincidences of four types (++, −−, +− and −+) counted by the coincidence monitor.
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