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 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 theorem is a complement to Bell's theorem (to be distinguished from the (Bell–)Kochen–Specker theorem of this article). 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 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 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 '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 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. 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 KS theorem explores whether it is possible to embed the set of quantum-mechanical observables into a set of classical quantities, in spite of the fact that all classical quantities are mutually compatible.The first observation made in the Kochen–Specker article is that this is possible in a trivial way, namely, by ignoring the algebraic structure of the set of quantum-mechanical observables. Indeed, let pA(ak) be the probability that observable A has value ak, then the product ΠA pA(ak), taken over all possible observables A, is a valid joint probability distribution, yielding all probabilities of quantum-mechanical observables by taking marginals. Kochen and Specker note that this joint probability distribution is not acceptable, however, since it ignores all correlations between the observables. Thus, in quantum mechanics A2 has value ak2 if A has value ak, implying that the values of A and A2 are highly correlated. More generally, it is required by Kochen and Specker that for an arbitrary function f the value v ( f ( A ) ) {displaystyle v{ig (}f(mathbf {A} ){ig )}} of observable f ( A ) {displaystyle f(mathbf {A} )} satisfies