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| In [[quantum mechanics]], the '''Kochen–Specker (KS) theorem'''<ref>S. Kochen and E.P. Specker, "The problem of hidden variables in quantum mechanics", ''Journal of Mathematics and Mechanics'' '''17''', 59–87 (1967).</ref> is a [[no-go theorem|"no go" theorem]]<ref>{{cite book |title=Interpreting the Quantum World |last=Bub |first=Jeffrey |authorlink=Jeffrey Bub |year=1999 |edition=revised paperback |publisher=Cambridge University Press |location= |isbn=978-0-521-65386-2 }}</ref> proved by [[Simon B. Kochen]] and [[Ernst Specker]] in 1967. It places certain constraints on the permissible types of [[Hidden variable theory|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]].
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| 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 [[Commutativity|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.
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| The Kochen–Specker proof demonstrates the impossibility of a version of Einstein's assumption, made in the famous [[EPR paradox|Einstein–Podolsky–Rosen]] paper,<ref>A. Einstein, B. Podolsky and N. Rosen, "Can quantum-mechanical description of physical reality be considered complete?" ''Phys. Rev.'' '''47''', 777–780 (1935).</ref> that quantum mechanical observables represent 'elements of physical reality'. More specifically, the theorem excludes [[Hidden variable theory|hidden variable theories]] that require elements of physical reality to be ''non''-contextual (i.e. independent of the measurement arrangement). As succinctly worded by [[Christopher Isham|Isham]] and [[Jeremy Butterfield|Butterfield]],<ref>[[Christopher Isham|C. J. Isham]], [[Jeremy Butterfield|J. Butterfield]], A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations, [http://arxiv.org/abs/quant-ph/9803055v4 arXiv:quant-ph/9803055v4] (submitted 20 March 1998, version of 13 October 1998)</ref> the Kochen–Specker theorem
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| : "asserts the impossibility of assigning values to all physical quantities whilst, at the same time, preserving the functional relations between them."
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| ==History==
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| 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 interpretation#Overview|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 [[EPR paradox|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<ref>N. Bohr, "Can quantum-mechanical description of physical reality be considered complete?" ''Phys. Rev.'' '''48''', 696–702 (1935).</ref> 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<ref>A. Einstein, "Quanten-Mechanik und Wirklichkeit", ''Dialectica'' '''2''', 320 (1948).</ref> 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.
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| 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 [[John von Neumann|von Neumann]],<ref>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.</ref> purporting to prove the impossibility of the hidden variable theories yielding the same results as quantum mechanics. First, [[David Bohm|Bohm]] developed an [[Bohm interpretation|interpretation of quantum mechanics]], generally accepted as a [[hidden variable theory]] underpinning quantum mechanics. The nonlocality of Bohm's theory induced [[John Stewart Bell|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's theorem|Bell inequality]], that is capable of being experimentally tested.
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| 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.
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| 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 [[David Mermin|Mermin]]<ref>N.D. Mermin, "What's wrong
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| 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).</ref> and by [[Asher Peres|Peres]].<ref>A. Peres, "Two simple proofs of the Kochen–Specker theorem", ''J. Phys. A: Math. Gen.'' '''24''', L175–L178 (1991).</ref> Many simpler proofs however only establish the theorem for Hilbert spaces of higher dimension, e.g., from dimension four.
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| ==The KS theorem==
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| The KS theorem explores whether it is possible to embed the set of quantum
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| mechanical observables into a set of ''classical'' quantities,
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| notwithstanding that all classical quantities are mutually compatible.
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| The first observation made in the Kochen–Specker paper is that this is possible in a trivial way, viz. by ignoring the algebraic structure of the set of quantum mechanical observables. Indeed, let ''p''<sub>'''A'''</sub>(''a''<sub>''k''</sub>) be the probability that observable '''A''' has value ''a''<sub>''k''</sub>, then the product Π<sub>'''A'''</sub>''p''<sub>'''A'''</sub>(''a''<sub>''k''</sub>), taken over all possible observables '''A''', is a valid [[joint probability distribution]], yielding all probabilities of quantum mechanical observables by taking [[Conditional probability|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 '''A'''<sup>2</sup> has value ''a''<sub>''k''</sub><sup>2</sup> if '''A''' has value ''a''<sub>''k''</sub>, implying that the values of '''A''' and '''A'''<sup>2</sup> are highly correlated.
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| More generally it is required by Kochen and Specker that for an arbitrary function f the value <math>\scriptstyle v(f({\mathbf A}))</math> of observable <math>\scriptstyle f({\mathbf A})</math> satisfies
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| :: <math>v(f({\mathbf A})) = f(v({\mathbf A})).</math>
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| If '''A'''<sub>1</sub> and '''A'''<sub>2</sub> are ''compatible'' (commeasurable) observables, then, by the same token, we should have the following two equalities
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| :: <math>v(c_1{\mathbf A}_1 + c_2{\mathbf A}_2) = c_1 v({\mathbf A}_1) + c_2 v({\mathbf A}_2),</math>
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| <math>c_1</math> and <math>c_2</math> real, and
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| :: <math>v({\mathbf A}_1{\mathbf A}_2) = v({\mathbf A}_1) v({\mathbf A}_2).</math>
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| The first of the latter two equalities is a considerable weakening compared to von Neumann's assumption that this equality should hold independently of whether '''A'''<sub>1</sub> and '''A'''<sub>2</sub> are compatible or incompatible. Kochen and Specker were capable of proving that a value assignment is not possible even on the basis of these weaker assumptions. In order to do so they restricted the observables to a special class, viz. so-called yes-no observables, having only values 0 and 1, corresponding to ''projection'' operators on the eigenvectors of certain orthogonal bases of a Hilbert space.
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| 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. Instead of the rather involved proof by Kochen and Specker it is more illuminating to reproduce here one of the much simpler proofs given much later, which employs a lower number of projection operators, but only proves the theorem when the dimension of the Hilbert space is at least 4. It turns out that it is possible to obtain a similar result on the basis of a set of only 18 projection operators.<ref>M. Kernaghan and A. Peres, Phys. Lett. A 198 (1995) 1–5.</ref>
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| In order to do so it is sufficient to realize that, if ''u''<sub>1</sub>, ''u''<sub>2</sub>, ''u''<sub>3</sub> and ''u''<sub>4</sub> are the four orthogonal vectors of an orthogonal basis in the four-dimensional Hilbert space, then the projection operators '''P'''<sub>1</sub>, '''P'''<sub>2</sub>, '''P'''<sub>3</sub>, '''P'''<sub>4</sub> on these vectors are all mutually commuting (and, hence, correspond to compatible observables, allowing a simultaneous attribution of values 0 or 1). Since
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| ::<math>{\mathbf P}_1+ {\mathbf P}_2+{\mathbf P_3}+ {\mathbf P_4} = {\mathbf I}</math>
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| it follows that
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| :: <math>v({\mathbf P_1}+ {\mathbf P_2}+{\mathbf P}_3+ {\mathbf P}_4) = v({\mathbf I}) = 1.</math>
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| But, since
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| :: <math>v({\mathbf P}_1+ {\mathbf P}_2+{\mathbf P}_3+ {\mathbf P}_4)=
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| v({\mathbf P}_1)+v({\mathbf P}_2)+v({\mathbf P}_3)+v({\mathbf P}_4)</math>
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| it follows from <math>\scriptstyle v({\mathbf P}_i) = </math> 0 or 1, <math>\scriptstyle i = 1,\ldots,4</math>, that out of the four values <math>\scriptstyle v({\mathbf P}_1), v({\mathbf P}_2), v({\mathbf P}_3), v({\mathbf P}_4)</math>, one must be 1 while the other three must be 0.
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| Cabello,<ref>A. Cabello, "A proof with 18 vectors of the Bell–Kochen–Specker theorem", in: M. Ferrero and A. van der Merwe (eds.), New Developments on Fundamental Problems in Quantum Physics, Kluwer Academic, Dordrecht, Holland, 1997, 59–62; Adan Cabello, Jose M. Estebaranz, Guillermo Garcia Alcaine, "Bell–Kochen–Specker theorem: A proof with 18 vectors", quant-ph/9706009v1, http://arxiv.org/abs/quant-ph/9706009v1</ref> extending an argument developed by Kernaghan <ref>M. Kernaghan, J. Phys. A 27 (1994) L829.</ref> considered 9 orthogonal bases, each basis corresponding to a column of the following table, in which the basis vectors are explicitly displayed. The bases are chosen in such a way that each has a vector in common with one other basis (indicated in the table by equal colours), thus establishing certain correlations between the 36 corresponding yes-no observables.
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| {| border="1" cellpadding="4" cellspacing="0"
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| |-
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| | ''u''<sub>1</sub>
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| | bgcolor="#FFFF00" | (0, 0, 0, 1)
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| | bgcolor="#FFFF00" | (0, 0, 0, 1)
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| | bgcolor="#00FFFF" | (1, –1, 1, –1)
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| | bgcolor="#00FFFF" | (1, –1, 1, –1)
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| | bgcolor="#AAFFAA" | (0, 0, 1, 0)
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| | bgcolor="#FF99FF" | (1, –1, –1, 1)
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| | bgcolor="#BBFFFF" | (1, 1, –1, 1)
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| | bgcolor="#BBFFFF" | (1, 1, –1, 1)
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| | bgcolor="#AAAACC" | (1, 1, 1, –1)
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| |-
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| | ''u''<sub>2</sub>
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| | bgcolor="#AAFFAA" | (0, 0, 1, 0)
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| | bgcolor="#CC44CC" | (0, 1, 0, 0)
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| | bgcolor="#FF99FF" | (1, –1, –1, 1)
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| | bgcolor="#DDDDAA" | (1, 1, 1, 1)
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| | bgcolor="#CC44CC" | (0, 1, 0, 0)
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| | bgcolor="#DDDDAA" | (1, 1, 1, 1)
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| | bgcolor="#AAAACC" | (1, 1, 1, –1)
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| | bgcolor="#EEDDFF" | (–1, 1, 1, 1)
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| | bgcolor="#EEDDFF" | (–1, 1, 1, 1)
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| |-
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| | ''u''<sub>3</sub>
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| | bgcolor="#FFFF88" | (1, 1, 0, 0)
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| | bgcolor="#7766AA" | (1, 0, 1, 0)
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| | bgcolor="#FFFF88" | (1, 1, 0, 0)
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| | bgcolor="#FFDDDD" | (1, 0, –1, 0)
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| | bgcolor="#DDFFDD" | (1, 0, 0, 1)
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| | bgcolor="#DD99BB" | (1, 0, 0, –1)
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| | bgcolor="#FF5555" | (1, –1, 0, 0)
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| | bgcolor="#7766AA" | (1, 0, 1, 0)
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| | bgcolor="#DDFFDD" | (1, 0, 0, 1)
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| |-
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| | ''u''<sub>4</sub>
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| | bgcolor="#FF5555" | (1, –1, 0, 0)
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| | bgcolor="#FFDDDD" | (1, 0, –1, 0)
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| | bgcolor="#55FF55" | (0, 0, 1, 1)
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| | bgcolor="#FF00FF" | (0, 1, 0, –1)
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| | bgcolor="#DD99BB" | (1, 0, 0, –1)
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| | bgcolor="#5555FF" | (0, 1, –1, 0)
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| | bgcolor="#55FF55" | (0, 0, 1, 1)
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| | bgcolor="#FF00FF" | (0, 1, 0, –1)
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| | bgcolor="#5555FF" | (0, 1, –1, 0)
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| |}
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| Now the "no go" theorem easily follows by making sure that it is impossible to
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| distribute the four numbers 1,0,0,0 over the four rows of each column, such that
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| equally coloured compartments contain equal numbers. Another way to see the theorem, using the approach by Kernaghan, is to recognize that a contradiction is implied between the odd number of bases and the even number of occurrences of the observables.
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| The usual proof of Bell's theorem ([[CHSH inequality]]) can also be converted into a simple proof of the KS theorem in dimension at least 4. Bell's setup involves four measurements with four outcomes (four pairs of a simultaneous binary measurement in each wing of the experiment) and four with two outcomes (the two binary measurements in each wing if the experiment, unaccompanied), thus 24 projection operators.
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| ==Remarks on the KS theorem==
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| 1. ''Contextuality''
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| In the Kochen–Specker paper the possibility is discussed that the value attribution <math>\scriptstyle v({\mathbf A})</math> 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 [[Homomorphism#Types of homomorphisms|homomorphic]] rather than isomorphic. This would make obsolete the requirement of a context-independent value attribution. Hence, the KS theorem does only exclude noncontextual hidden variable theories. The possibility of contextuality has given rise to the so-called [[Interpretation of quantum mechanics#Modal interpretations of quantum theory|modal interpretations of quantum mechanics]].
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| 2. ''Different levels of description''
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| 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 question may be asked whether this is a very shocking result. 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
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| rather than quantum mechanics. In later publications<ref>e.g. J.F. Clauser and M.A. Horne, "Experimental consequences of objective local theories", ''Physical Review D'' '''10''', 526–535 (1974).</ref> 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, incidentally, 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. The existence or nonexistence of such ''subquantum'' elements of physical reality is not touched by the KS theorem. As an example, recent experiments on bouncing drops on a vibrating bath, by Y. Couder and collaborators, reproduce many features of quantum mechanics.<ref>John W. M. Bush, Quantum mechanics writ large, ''PNAS'' '''107''', 17455–17456 (2010). [http://www.pnas.org/content/107/41/17455.short]</ref><ref>Protière S., Boudaoud A., Couder Y. Particle wave association on a fluid interface ''J. Fluid Mech.'' '''554''', 85-108 (2006).</ref><ref>Couder Y., Protière S., Fort E., Boudaoud A. Dynamical phenomena: Walking and orbiting droplets
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| ''Nature'' '''437''', 208 (2005).</ref><ref>Couder Y., Fort E. Single-particle diffraction and interference at a macroscopic scale ''Phys. Rev. Lett.'' '''97''', 154101-1–154101-4 (2006).</ref><ref>Eddi A., Fort E., Moisy F., Couder Y. Unpredictable tunneling of a classical wave-particle association ''Phys. Rev. Lett.'' '''102''', 240401-1–240401-4 (2009).</ref><ref>Eddi A., Decelle A., Fort E., Couder Y. Archimedean lattices in the bound states of wave interacting particles ''Europhys. Lett.'' '''87''', 56002-p1–56002-p6 (2009).</ref><ref>Fort E., Eddi A., Boudaoud A., Moukhtar J., Couder Y. Path-memory induced quantization of classical orbits ''Proc. Natl. Acad. Sci. USA'' '''107''', 17515–17520 (2010).</ref> In this case, the ''subquantum'' elements of physical reality are linked to the specifics of the hydrodynamics of bouncing drops on a vibrating bath (linked to the [[Faraday wave]] instability phenomenology <ref>Couder Y., Fort E., Gautier C.H., Boudaoud A. From bouncing to floating: Noncoalescence of drops on a fluid bath ''Phys. Rev. Lett.'' '''94''' 177801-1–177801-4 (2005).</ref>). At the level that reproduces features of quantum mechanics, measurement are not ''deterministic'' since they depend on the [[stochastic]] nature of the [[non-linear dynamics]] of the ''subquantum'' elements. The experiments are indeed interpreted in the framework of [[De Broglie–Bohm theory]] of pilot waves.
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| ==Notes==
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| <references/>
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| == External links ==
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| *Carsten Held, ''The Kochen–Specker Theorem'', Stanford Encyclopedia of Philosophy *[http://plato.stanford.edu/entries/kochen-specker/]
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| *[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=-379340 Kochen–Specker theorem on arxiv.org]
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| *S. Kochen and E. P. Specker, The problem of hidden variables in quantum mechanics, Full text [http://www.iumj.indiana.edu/IUMJ/dfulltext.php?year=1968&volume=17&artid=17004]
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| {{DEFAULTSORT:Kochen-Specker theorem}}
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| [[Category:Quantum mechanics]]
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| [[Category:Hidden variable theory]]
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| [[Category:Theorems in quantum physics]]
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