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In quantum mechanics, the Kochen–Specker (KS) theorem〔S. Kochen and E.P. Specker, "The problem of hidden variables in quantum mechanics", ''Journal of Mathematics and Mechanics'' 17, 59–87 (1967).〕 is a "no go" theorem proved 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 theorem is a complement to Bell's theorem. 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 of a version of Einstein's assumption, made in the famous Einstein–Podolsky–Rosen paper,〔A. Einstein, B. Podolsky and N. Rosen, "Can quantum-mechanical description of physical reality be considered complete?" ''Phys. Rev.'' 47, 777–780 (1935).〕 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,〔C. J. Isham, J. Butterfield, A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations, (arXiv:quant-ph/9803055v4 ) (submitted 20 March 1998, version of 13 October 1998)〕 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." ==History== 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〔N. Bohr, "Can quantum-mechanical description of physical reality be considered complete?" ''Phys. Rev.'' 48, 696–702 (1935).〕 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〔A. Einstein, "Quanten-Mechanik und Wirklichkeit", ''Dialectica'' 2, 320 (1948).〕 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 '60s two lines of development were open for those not averse to metaphysics, both lines improving on a "no go" theorem presented by von Neumann,〔J. von Neumann, ''Mathematische Grundlagen der Quantenmechanik'', Springer, Berlin, 1932; English translation: ''Mathematical foundations of quantum mechanics'', Princeton Univ. Press, 1955, Chapter IV.1,2.〕 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 ''non''local, 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 ''in''compatibility 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 also published a proof of the Kochen–Specker theorem in 1967, in a paper which had been submitted to a journal earlier than his famous Bell-inequality paper, 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〔N.D. Mermin, "What's wrong with these elements of reality?" ''Physics Today'', 43, Issue 6, 9–11 (1990); N.D. Mermin, "Simple unified form for the major no-hidden-variables theorems", ''Phys. Rev. Lett.'' 65, 3373 (1990).〕 and by Peres.〔A. Peres, "Two simple proofs of the Kochen–Specker theorem", ''J. Phys. A: Math. Gen.'' 24, L175–L178 (1991).〕 Many simpler proofs however only establish the theorem for Hilbert spaces of higher dimension, e.g., from dimension four. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kochen–Specker theorem」の詳細全文を読む スポンサード リンク
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