Home > Bell's theorem
Bell's theorem was first proposed in 1964 by the physicist John S. Bell, and provides the basis for experimental tests of local realism versus quantum mechanics. It states that a certain inequality has to be satisfied if the world works according to the principle of "local realism" but can be violated by quantum mechanics. The history and implications of the theorem are discussed in the EPR paradox and interpretation of quantum mechanics pages, and in many popular books, some of which are listed at the end of this article.Although the theorem is named after John Bell, he himself never formulated it as such and, since publication of his paper in 1964, a number of different inequalities have been derived, all termed "Bell inequalities". They all purport to make the same assumptions about local realism -- that even quantum-level objects have well defined parameters and that distant objects do not exchange information instantaneously -- though some in fact make additional assumptions and so are less general than others. The inequalities concern measurements made on pairs of particles that have interacted and then separated. They place limits on the possible values of the " quantum correlation " that can be obtained, on the assumption that the only possible factors causing the correlation are shared "hidden variables", set at the souce -- the point of interaction. In quantum mechanics, the existence of these hidden variables associated with individual particle pairs is denied, the correlation being instead due to quantum entanglement of the whole ensemble. The correlation is predicted to exceed Bell's limits.
" Bell test experiments" to date suggest that Bell's inequality is violated, and the
general belief in the physics community is that they prove quantum mechanics is correct and local realism does not hold. Nevertheless, the issue is far from being conclusively settled. The experiments themselves can be challenged on the grounds that they make various assumptions which open "loopholes" and may have led to misinterpretation. Designing experiments which more conclusively determine whether or not the Bell inequality holds remains a very active area of research (Shimony, 2004), as does the search for local hidden variable theories that can explain existing Bell test violations.
1 Bell test experiments
If the world has underlying local hidden variables, then Bell's inequalities must be obeyed by the "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. Several different subexperiments are conducted, one set for each of up to four choices of setting. In some Bell tests the analysers only have a single output channel. For these, extra subexperiments are needed in which polarisers are absent on one or other or both sides, and a Bell inequality designed for the purpose (the CH74 inequality) is used.
2 The local hidden variable assumption
2.1 Existence of hidden variables
Under local realism, it is assumed that the photons in a Bell test experiment have definite properties such as polarisation direction even before they are detected. This contrasts with quantum mechanics, which says that the particles emitted from a quantum event exist in a superposition of polarisation states until detected. Bell's original paper concerned "spin-½" particles rather than photons, and he assumed each particle had definite spin, the "hidden variable" λ. He assumed that knowing λ and the detector setting, the outcome was completely determined.
2.2 Stochastic hidden variables
It was later found (Bell, 1971; Clauser, 1974) that in real experiments it was best to recognise the fact that the components of λ (and there can be many) fall into two sets, those that describe the quantum state of the source at the moment of emission and "others", primarily associated with the polarisers and detectors. It is the former that play the logical role of hidden variables, the latter being effectively random. They can be averaged out, leaving the non-random ones as "stochastic hidden variables", determining (together with the polariser settings) not the actual outcomes but just their probabilities. Clauser and Horne termed a theory based on this kind of hidden variable an "Objective Local Theory", or OLT. It is this kind that is used in the later (definitive) derivations of the inequalities used in actual experiments.
2.3 Locality
The other vital assumption is "locality" -- once the particles are separated they do not have any influence on each other, and the two detectors do not affect each other. Only "local" factors can affect each particle. In some experiments elaborate precautions have been taken to ensure that if there were any influence of one side of the experiment on the other then it could only be one that acted faster than the speed of light.
