Kochen–Specker theorem
[4] 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.[1] 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 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,[5] (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".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.Taking into account the contextuality stemming from the measurement arrangement would, according to Bohr, make invalid the EPR reasoning.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.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.The first experimental test of contextuality was performed in 2000,[12] and a version without detection, sharpness and compatibility loopholes was achieved in 2022.real, and The first of these is a considerable weakening compared to von Neumann's assumption that this equality should hold independently of whether A1 and A2 are compatible or incompatible.As long as the Hilbert space is at least three-dimensional, they were able to find a set of 117 such projection operators, not allowing to attribute to each of them in an unambiguous way either value 0 or 1.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.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.In later publications[18] 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.