🔗 Kochen–Specker theorem

🔗 Physics

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). While Bell's theorem established nonlocality to be a feature of any hidden variable theory that recovers the predictions of quantum mechanics, the KS theorem established contextuality to be an inevitable feature of such theories.

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 theorem excludes hidden-variable theories that assume that elements of physical reality can all be consistently represented simultaneously by the quantum mechanical Hilbert space formalism disregarding the context of a particular framework (technically a projective decomposition of the identity operator) related to the experiment or analytical viewpoint under consideration. As succinctly worded by Isham and Butterfield, (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".