Kochen–Specker theorem

In quantum mechanics, the Kochen–Specker (KS) theorem,^[1] also known as the Bell-Kochen–Specker theorem,^[2] is a "no go" theorem^[3] 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 apparent randomness of quantum mechanics as a deterministic model featuring hidden states. 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).

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 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,^[4] 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^[5] 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^[6] 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,^[7] 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 nonlocal, 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 incompatibility 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^[8] and by Peres.^[9] Many simpler proofs however only establish the theorem for Hilbert spaces of higher dimension, e.g., from dimension four.

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, in spite of the fact 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 $\scriptstyle v(f({\mathbf A}))$ of observable $\scriptstyle f({\mathbf A})$ 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.

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.^[10]

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 $\scriptstyle v({\mathbf P}_i) =$ 0 or 1, $\scriptstyle i = 1,\ldots,4$, that out of the four values $\scriptstyle v({\mathbf P}_1), v({\mathbf P}_2), v({\mathbf P}_3), v({\mathbf P}_4)$, one must be 1 while the other three must be 0.

Cabello,^[11] extending an argument developed by Kernaghan ^[12] 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.

u1	(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)
u2	(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)
u3	(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)
u4	(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. 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.

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 of the experiment, unaccompanied), thus 24 projection operators.

Remarks on the KS theorem

1. Contextuality

In the Kochen–Specker paper the possibility is discussed that the value attribution $\scriptstyle v({\mathbf A})$ 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, it 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 only excludes 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 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. In later publications^[13] 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 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.

Notes

1. S. Kochen and E.P. Specker, "The problem of hidden variables in quantum mechanics", Journal of Mathematics and Mechanics 17, 59–87 (1967).
2. J.S.Bell, "On the problem of hidden variables in quantum mechanics", Reviews of Modern Physics 38, 447-452 (1966).
3. Bub, Jeffrey (1999). Interpreting the Quantum World (revised paperback ed.). Cambridge University Press. ISBN 978-0-521-65386-2.
4. 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)
5. N. Bohr, "Can quantum-mechanical description of physical reality be considered complete?" Phys. Rev. 48, 696–702 (1935).
6. A. Einstein, "Quanten-Mechanik und Wirklichkeit", Dialectica 2, 320 (1948).
7. 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.
8. 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).
9. A. Peres, "Two simple proofs of the Kochen–Specker theorem", J. Phys. A: Math. Gen. 24, L175–L178 (1991).
10. M. Kernaghan and A. Peres, Phys. Lett. A 198 (1995) 1–5.
11. 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
12. M. Kernaghan, J. Phys. A 27 (1994) L829.
13. e.g. J.F. Clauser and M.A. Horne, "Experimental consequences of objective local theories", Physical Review D 10, 526–535 (1974).

External links

• Carsten Held, The Kochen–Specker Theorem, Stanford Encyclopedia of Philosophy *
• Kochen–Specker theorem on arxiv.org
• S. Kochen and E. P. Specker, The problem of hidden variables in quantum mechanics, Full text