Kochen-Specker theorem

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In quantum mechanics, the Kochen-Specker (KS) theorem^[1] is a "no go" theorem proved by Simon 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 theory 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 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 generated by the fact that quantum mechanical observables need not be commutative, making it impossible to embed the algebra of these observables in a commutative algebra, by assumption representing the classical structure of the hidden variables theory.

The Kochen-Specker proof demonstrates the impossibility of Einstein's assumption, made in the famous Einstein-Podolsky-Rosen paper^[2], that quantum mechanical observables represent `elements of physical reality'. More generally does the theorem exclude hidden variable theories requiring elements of physical reality to be noncontextual (i.e. independent of the measurement arrangement).

1 History
2 The KS theorem
3 Remarks on the KS theorem
4 References
5 External links

[edit] 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 before the measurement to be present 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 presupposition the EPR criticism was not taken very seriously by the majority of the physics community. Moreover, in his answer^[3] to the EPR paper Bohr had pointed to an ambiguity in the EPR paper, to the effect that it is assumed there that the value of a quantum mechanical observable is non-contextual (i.e. 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^[4] that Bohr's reliance on contextuality implies nonlocality, and that, as a consequence, one would have to accept incompleteness if one wanted to avoid nonlocality (spooky action at a distance).

In the 1950's and 60's two lines of development were open for those not being averse of metaphysics, both lines improving on a "no go" theorem presented by von Neumann^[5], purporting to prove the impossibility of hidden variable theories yielding the same results as does 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 did Bell manage to lift the problem from the metaphysical level to the physical one by deriving an inequality, the Bell inequality, that is liable to be experimentally tested.

A second line is the Kochen-Specker one. The essential difference with Bell's approach is that the possibility of an underpinning of quantum mechanics by a hidden variable theory is dealt with independently of any reference to nonlocality. Although contextuality plays an important part also here, there is no implication of nonlocality because the proof refers to observables belonging to one single object, to be measured in one and the same region of space. Contextuality is related here with incompatibility of quantum mechanical observables, incompatibility being associated with mutual exclusiveness of measurement arrangements.

Considerably simpler proofs than the Kochen-Specker one were given later, amongst others, by Mermin^[6] and by Peres^[7].

[edit] The KS theorem

The KS theorem explores whether it is possible to embed the set of quantum mechanical observables into a set of classical quantities, notwithstanding that all classical quantities are mutually compatible. 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_A(a_k) be the probability that observable A has value a_k, then the product $Π$ _Ap_A(a_k), 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 a_k² if A has value a_k, 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)) of observable f(A) satisfies

$v(f({\mathbf A})) = f(v({\mathbf A}))$

If A₁ and A₂ are compatible (commeasurable) observables, then, by the same token, we should have the following two equalities

$v(c_1{\mathbf A}_1 + c_2{\mathbf A}_2) = c_1 v({\mathbf A}_1) + c_2 v({\mathbf A}_2),$

$c 1$ and $c 2$ real, and

$v({\mathbf A}_1{\mathbf A}_2) = v({\mathbf A}_1) v({\mathbf A}_2)$

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 A₁ and A₂ 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.

Restricting themselves to a three-dimensional Hilbert space, 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 allow to employ a much lower number of projection operators by considering a four-dimensional Hilbert space. It turns out that it is possible to obtain a similar result on the basis of a set of only 18 projection operators.

In order to do so it is sufficient to realize that, if u₁, u₂, u₃ and u₄ are the four orthogonal vectors of an orthogonal basis in the four-dimensional Hilbert space, then the projection operators P₁, P₂, P₃, P₄ on these vectors are all mutually commuting (and, hence, correspond to compatible observables, allowing a simultaneous attribution of values 0 or 1). Since

${\mathbf P}_1+ {\mathbf P}_2+{\mathbf P_3}+ {\mathbf P_4} = {\mathbf I}$

it follows that

$v({\mathbf P_1}+ {\mathbf P_2}+{\mathbf P}_3+ {\mathbf P}_4) = v({\mathbf I}) = 1$

But, since

$v({\mathbf P}_1+ {\mathbf P}_2+{\mathbf P}_3+ {\mathbf P}_4)= v({\mathbf P}_1)+v({\mathbf P}_2)+v({\mathbf P}_3)+v({\mathbf P}_4)$

it follows from v(P_i) = 0 or 1, i = 1...4, that out of the four values v(P₁), v(P₂), v(P₃), v(P₄) one must be 1 while the other three must be 0.

By Cabello^[8] 9 orthogonal bases were considered, 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.

$u 1$	(0,0,0,1)	(0,0,0,1)	(1,-1,1,-1)	(1,-1,1,-1)	(0,0,1,0)	(1,-1,-1,1)	(1,1,-1,1)	(1,1,-1,1)	(1,1,1,-1)
$u 2$	(0,0,1,0)	(0,1,0,0)	(1,-1,-1,1)	(1,1,1,1)	(0,1,0,0)	(1,1,1,1)	(1,1,1,-1)	(-1,1,1,1)	(-1,1,1,1)
$u 3$	(1,1,0,0)	(1,0,1,0)	(1,1,0,0)	(1,0,-1,0)	(1,0,0,1)	(1,0,0,-1)	(1,-1,0,0)	(1,0,1,0)	(1,0,0,1)
$u 4$	(1,-1,0,0)	(1,0,-1,0)	(0,0,1,1)	(0,1,0,-1)	(1,0,0,-1)	(0,1,-1,0)	(0,0,1,1)	(0,1,0,-1)	(0,1,-1,0)

Now the "no go" theorem easily follows by making sure that it is impossible to distribute the four numbers 1,0,0,0 over the four rows of each column, such that equally coloured compartments contain equal numbers.

[edit] Remarks on the KS theorem

1. Contextuality

In the Kochen-Specker paper the possibility is discussed that the value attribution $v({\mathbf A})$ 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, i t is possible that relations between quantum mechanical observables and hidden variables are just 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 modal interpretations of quantum mechanics.

2. Different levels of description

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 rather than quantum mechanics. Indeed, in later publications^[9] 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.

[edit] References

^ S. Kochen and E.P. Specker, "The problem of hidden variables in quantum mechanics", Journal of Mathematics and Mechanics 17, 59-87 (1967).
^ A. Einstein, B. Podolsky and N. Rosen, "Can quantum-mechanical description of physical reality be considered complete?" Phys. Rev. 47, 777-780 (1935).
^ N. Bohr, "Can quantum-mechanical description of physical reality be considered complete?" Phys. Rev. 48, 696-702 (1935).
^ A. Einstein, "Quanten-Mechanik und Wirklichkeit", Dialectica 2, 320 (1948).
^ 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.
^ 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).
^ A. Peres, "Two simple proofs of the Kochen-Specker theorem", J. Phys. A: Math. Gen. 24, L175-L178 (1991).
^ 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://xxx.lanl.gov/PS_cache/quant-ph/pdf/9706/9706009v1.pdf
^ e.g. J.F. Clauser and M.A. Horne, "Experimental consequences of objective local theories", Physical Review D 10, 526-535 (1974).