Теорема Кохена — Шпекера и ограничения теорий скрытых переменных.

Theorem constraining types of hidden variable theories
In quantum mechanics, the Kochen–Specker (KS) theorem, also known as the Bell–KS 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 predictions of quantum mechanics in a context independent way. 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 Kochen–Specker theorem is a complement to Bell's theorem. While Bell's theorem established nonlocality to be a feature of any hidden variable theory that recovers the predictions of quantum mechanics, the Kochen–Specker theorem established contextuality to be an inevitable feature of such theories. (under the assumption of a universal probabilistic sample space as in non contextual hidden variable theories) the Kochen–Specker theorem "asserts the impossibility of assigning values to all physical quantities whilst, at the same time, preserving the functional relations between them".

The KS theorem is an important step in the debate on the (in)completeness of quantum mechanics, boosted in 1935 by the criticism of the Copenhagen assumption of completeness in the article by Einstein, Podolsky and Rosen, 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 article 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 article, to the effect that it assumes that you can suppose nothing would have changed in the distant results of the measurements changing the local measurement basis, even if all the universal context was different. Taking into account the contextuality stemming from the measurement arrangement would, according to Bohr, make invalid 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. In the 1950s and 1960s two lines of development were open for those not averse to metaphysics, both lines improving on a "no go" theorem presented by von Neumann, purporting to prove the impossibility of the hidden variable theories yielding the same results as quantum mechanics. First, Bohm developed an interpretation of quantum mechanics, generally accepted as a hidden variable theory underpinning quantum mechanics. The nonlocality of Bohm's theory induced Bell to assume that quantum reality is nonlocal, and that probably only local hidden variable theories are in disagreement with quantum mechanics. More importantly, Bell managed to lift the problem from the level of metaphysics to physics by deriving an inequality, the Bell inequality, that is capable of being experimentally tested. A second line is the Kochen–Specker one. The essential difference from Bell's approach is that the possibility of underpinning quantum mechanics by a hidden variable theory is dealt with independently of any reference to locality or nonlocality, but instead a stronger restriction than locality is made, namely that hidden variables are exclusively associated with the quantum system being measured; none are associated with the measurement apparatus. This is called the assumption of non contextuality. Contextuality is related here with incompatibility of quantum mechanical observables, incompatibility being associated with mutual exclusiveness of measurement arrangements. The Kochen–Specker theorem states that no non contextual hidden variable model can reproduce the predictions of quantum theory when the dimension of the Hilbert space is three or more. Bell published a proof of the Kochen–Specker theorem in 1966, in an article which had been submitted to a journal earlier than his famous Bell inequality article, but was lost on an editor's desk for two years. Considerably simpler proofs than the Kochen–Specker one were given later, amongst others, by Mermin and by Peres. However, many simpler proofs only establish the theorem for Hilbert spaces of higher dimension, e. g., from dimension four. The first experimental test of contextuality was performed in 2000, and a version without detection, sharpness and compatibility loopholes was achieved in 2022.

In the Kochen–Specker article the possibility is discussed that the value attribution may be context dependent, i. e. observables corresponding to equal vectors in different columns of the table need not have equal values because different columns correspond to different measurement arrangements. Since subquantum reality (as described by the hidden variable theory) may be dependent on the measurement context, it is possible that relations between quantum mechanical observables and hidden variables are just homomorphic rather than isomorphic. This would make obsolete the requirement of a context independent value attribution. Hence, the KS theorem only excludes noncontextual hidden variable theories. The possibility of contextuality has given rise to the so called modal interpretations of quantum mechanics.

By the KS theorem the impossibility is proven of Einstein's assumption that an element of physical reality is represented by a value of a quantum mechanical observable. The value of a quantum mechanical observable refers in the first place to the final position of the pointer of a measuring instrument, which comes into being only during the measurement, and which, for this reason, cannot play the role of an element of physical reality. Elements of physical reality, if existing, would seem to need a subquantum (hidden variable) theory for their description rather than quantum mechanics. In later publications the Bell inequalities are discussed on the basis of hidden variable theories in which the hidden variable is supposed to refer to a subquantum property of the microscopic object different from the value of a quantum mechanical observable. This opens up the possibility of distinguishing different levels of reality described by different theories, which had already been practised by Louis de Broglie. For such more general theories the KS theorem is applicable only if the measurement is assumed to be a faithful one, in the sense that there is a deterministic relation between a subquantum element of physical reality and the value of the observable found on measurement.