Foundations of Physics

, Volume 44, Issue 3, pp 248–265 | Cite as

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

  • Ehtibar N. DzhafarovEmail author
  • Janne V. Kujala


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.


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



This work was supported by NSF Grant SES-1155956.


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Purdue UniversityWest LafayetteUSA
  2. 2.Department of Mathematical Information TechnologyUniversity of JyväskyläJyväskyläFinland

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