Unification of Bell, Leggett-Garg and Kochen-Specker inequalities: Hybrid spatio-temporal inequalities

¹ Indian Institute of Science Education and Research - Pune-411008, India
² Poornaprajna Institute of Scientific Research - Sadashivnagar, Bengaluru-560080, India
³ Raman Research Institute - Sadashivnagar, Bengaluru-560060, India
⁴ CAPSS, Bose Institute - Kolkata-700 091, India

^(a) sdas@students.iiserpune.ac.in
^(b) aru@poornaprajna.org
^(c) srik@poornaprajna.org
^(d) dhome@jcbose.ac.in

Received: 26 August 2013
Accepted: 2 January 2014

Abstract

The Bell-type (spatial), Kochen-Specker (contextuality) or Leggett-Garg (temporal) inequalities are based on classically plausible but otherwise quite distinct assumptions. For any of these inequalities, satisfaction is equivalent to a joint probability distribution for all observables in the experiment. This implies a joint distribution for all pairs of observables, and is indifferent to whether or not they commute in the theory. This indifference underpins a unification of the above inequalities into a general framework of correlation inequalities. When the physical scenario is such that the correlated pairs are all compatible, the resulting correlation is nonsignaling, which may be local or multi-particle, corresponding to contextuality or Bell-type inequalities. If the pairs are incompatible, the resulting correlation corresponds to Leggett-Garg (LG) inequalities. That quantum mechanics (QM) violates all these inequalities suggests a close connection between the local, spatial and temporal properties of the theory. As a concrete manifestation of the unification, we extend the method due to Roy and Singh (J. Phys. A, 11 (1978) L167) to derive and study a new class of hybrid spatio-temporal inequalities, where the correlated pairs in the experiment are both compatible or incompatible. The implications for cryptography and monogamy inequalities of the unification are briefly touched upon.