No-Forcing and No-Matching Theorems for Classical Probability Applied to Quantum Mechanics

Abstract

Correlations of spins in a system of entangled particles are inconsistent with Kolmogorov's probability theory (KPT), provided the system is assumed to be non-contextual. In the Alice-Bob EPR paradigm, non-contextuality means that the identity of Alice's spin (i.e., the probability space on which it is defined as a random variable) is determined only by the axis \alphai chosen by Alice, irrespective of Bob's axis \betaj (and vice versa). Here, we study contextual KPT models, with two properties: (1) Alice's and Bob's spins are identified as Aij and Bij, even though their distributions are determined by, respectively, \alphai alone and \betaj alone, in accordance with the no-signaling requirement; and (2) the joint distributions of the spins Aij,Bij across all values of \alphai,\betaj are constrained by fixing distributions of some subsets thereof. Of special interest among these subsets is the set of probabilistic connections, defined as the pairs \left(Aij,Aij'\right) and \left(Bij,Bi'j\right) with \alphai\not=\alphai' and \betaj\not=\betaj' (the non-contextuality assumption is obtained as a special case of connections, with zero probabilities of Aij\not=Aij' and Bij\not=Bi'j). Thus, one can achieve a complete KPT characterization bof the Bell-type inequalities, or Tsirelson's inequalities, by specifying the distributions of probabilistic connections compatible with those and only those spin pairs \left(Aij,Bij\right) that are subject to these inequalities. We show, however, that quantum-mechanical (QM) constraints are special. No-forcing theorem says that if a set of probabilistic connections is not compatible with correlations violating QM, then it is compatible only with the classical-mechanical correlations. No-matching theorem says that there are no subsets of the spin variables Aij,Bij whose distributions can be fixed to be compatible with and only with QM-compliant correlations.