Foundations of Physics

, Volume 44, Issue 3, pp 248–265

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

Article

## 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 $$\alpha _{i}$$ chosen by Alice, irrespective of Bob’s axis $$\beta _{j}$$ (and vice versa). Here, we study contextual KPT models, with two properties: (1) Alice’s and Bob’s spins are identified as $$A_{ij}$$ and $$B_{ij}$$, even though their distributions are determined by, respectively, $$\alpha _{i}$$ alone and $$\beta _{j}$$ alone, in accordance with the no-signaling requirement; and (2) the joint distributions of the spins $$A_{ij},B_{ij}$$ across all values of $$\alpha _{i},\beta _{j}$$ 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 $$\left( A_{ij},A_{ij'}\right)$$ and $$\left( B_{ij},B_{i'j}\right)$$ with $$\alpha _{i}\not =\alpha _{i'}$$ and $$\beta _{j}\not =\beta _{j'}$$ (the non-contextuality assumption is obtained as a special case of connections, with zero probabilities of $$A_{ij}\not =A_{ij'}$$ and $$B_{ij}\not =B_{i'j}$$). Thus, one can achieve a complete KPT characterization of the Bell-type inequalities, or Tsirelson’s inequalities, by specifying the distributions of probabilistic connections compatible with those and only those spin pairs $$\left( A_{ij},B_{ij}\right)$$ 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 $$A_{ij},B_{ij}$$ whose distributions can be fixed to be compatible with and only with QM-compliant correlations.

## Keywords

CHSH inequalities Contextuality EPR/Bohm paradigm  Probabilistic couplings Random variables Tsirelson inequalities

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