On the range of non-classical probability

Abstract

A non-classical probability is defined in the framework of the generalized probability theory as a probability measure on the events represented as an orthomodular lattice admitting a full set of probability measures. This notion generalizes the classical probability as well as the Hilbert space probability used in quantum mechanics. In this paper it is shown that the range of non-classical correlation sequences of the form p = ( p 1,…, p n , p 12, p 23,…, p n − 1, n , p 1 n ) is closed and convex and coincides with the polytope defined by the inequalities 0 ≤ p i ≤ 1,0 ≤ p ij ≤ min{ p i , p j }. This generalizes the theorem of Pitowsky who showed that for Hilbert space quantum mechanics the range of p is convex but not closed. Since the cases n = 3 and n = 4 correspond to the situation with Bell's and Clauser-Horne inequalities, the result of the paper shows that there are no counterparts of Bell-type inequalities for non-classical probabilities, except for the inequalities mentioned above.