Quantum analogues of the Bell inequalities. The case of two spatially separated domains

Abstract

THE CASE OF UDC 519.2 One investigates inequalities for the probabilities and mathematical expectations which follow from the postulates of the local quantum theory. It turns out that the relation between the quantum and the classical correlation matrices is expressed in terms of Grothendieck's known constant. It-is also shown that the extremal quantum correlations characterize the Clifford algebra (i.e., canonical anticommutative relations).The Bell inequalities are inequalities for probabilities that are valid in any local deterministic theory with hidden parameters (briefly and not entirely exactly, these theories will be said to be classical), but need not be true in quantum theory; see, for example, the surveys [1-4], and also [5, pp. 190-193], and [6]. The domain of the probability distributions, admissible in the classical theories, lends itself to a mathematical description [7] and this description is model-independent, i.e., it is not connected with any concrete physical mechanisms.On the other hand, for the probability distributions, admissible in the quantum theory, one considers usually only certain special cases; moreover, in the mentioned paper [7] one has expressed scepticism regarding the possibility of a model-independent approach to quantum probabilities. However, such an approach is possible (it has been communicated by the author in [9]) for a very general situation, allowing many domains in space--time~ both in spatially separated domains is considered in more detail in this paper; one proves certain theorems, regarding this case, which have been communicated in [9]. Then we carry out a comparison of the quantum case with the classical one; here, unexpectedly, there arises the Grothendieck constant, known from the geometry of Banach spaces. It turns out that the quantum correlations exceed the classical ones at most 1782 times. In the last section we present some preliminary results for the case of three spatially separated domains. i. Some Facts about Clifford AlgebrasThis auxiliary section prepares the technical means, used in the subsequent sect:ions.By a Clifford algebra 0(~) we mean a C*-algebra, generated by the Hermitian genera-

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