Kochen–Specker theorem

In quantum mechanics, the Kochen–Specker (KS) theorem, also known as the Bell–Kochen–Specker theorem, is a "no go" theorem proved by John S. Bell in 1966 and by Simon B. Kochen and Ernst Specker in 1967. It places certain constraints on the permissible types of hidden variable theories which try to explain the apparent randomness of quantum mechanics as a deterministic model featuring hidden states. The version of the theorem proved by Kochen and Specker also gave an explicit example for this constraint in terms of a finite number of state vectors. The theorem is a complement to Bell's theorem (to be distinguished from the (Bell–)Kochen–Specker theorem of this article).

The theorem proves that there is a contradiction between two basic assumptions of the hidden variable theories intended to reproduce the results of quantum mechanics: that all hidden variables corresponding to quantum mechanical observables have definite values at any given time, and that the values of those variables are intrinsic and independent of the device used to measure them. The contradiction is caused by the fact that quantum mechanical observables need not be commutative. It turns out to be impossible to simultaneously embed all the commuting subalgebras of the algebra of these observables in one commutative algebra, assumed to represent the classical structure of the hidden variables theory, if the Hilbert space dimension is at least three.

The Kochen–Specker proof demonstrates the impossibility that quantum mechanical observables represent "elements of physical reality". More specifically, the theorem excludes hidden variable theories that require elements of physical reality to be non-contextual (i.e. independent of the measurement arrangement). As succinctly worded by Isham and Butterfield, the Kochen–Specker theorem

The KS theorem is an important step in the debate on the (in)completeness of quantum mechanics, boosted in 1935 by the criticism in the EPR paper of the Copenhagen assumption of completeness, creating the so-called EPR paradox. This paradox is derived from the assumption that a quantum mechanical measurement result is generated in a deterministic way as a consequence of the existence of an element of physical reality assumed to be present before the measurement as a property of the microscopic object. In the EPR paper it was assumed that the measured value of a quantum mechanical observable can play the role of such an element of physical reality. As a consequence of this metaphysical supposition the EPR criticism was not taken very seriously by the majority of the physics community. Moreover, in his answer Bohr had pointed to an ambiguity in the EPR paper, to the effect that it assumes the value of a quantum mechanical observable is non-contextual (i.e. is independent of the measurement arrangement). Taking into account the contextuality stemming from the measurement arrangement would, according to Bohr, make obsolete the EPR reasoning. It was subsequently observed by Einstein that Bohr's reliance on contextuality implies nonlocality ("spooky action at a distance"), and that, in consequence, one would have to accept incompleteness if one wanted to avoid nonlocality.

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