Bell inequalities in four dimensional phase space and the three marginal theorem

Auberson, G. ; Mahoux, G. ; Roy, S. M. ; Singh, Virendra (2003) Bell inequalities in four dimensional phase space and the three marginal theorem Journal of Mathematical Physics, 44 (7). pp. 2729-2748. ISSN 0022-2488

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Official URL: http://link.aip.org/link/?JMAPAQ/44/2729/1

Related URL: http://dx.doi.org/10.1063/1.1578532

Abstract

We address the classical and quantum marginal problems, namely the question of simultaneous realizability through a common probability density in phase space of a given set of compatible probability distributions. We consider only distributions authorized by quantum mechanics, i.e., those corresponding to complete commuting sets of observables. For four-dimensional phase space with position variables q and momentum variables p, we establish the two following points: (i) given four compatible probabilities for (q1,q2), (q1,p2), (p1,q2), and (p1,p2), there does not always exist a positive phase space density ρ(q,p) reproducing them as marginals; this settles a long standing conjecture; it is achieved by first deriving Bell-type inequalities in phase space which have their own theoretical and experimental interest. (ii) Given instead at most three compatible probabilities, there always exist an associated phase space density ρ(q,p); the solution is not unique and its general form is worked out. These two points constitute our "three marginal theorem."

Item Type:Article
Source:Copyright of this article belongs to American Institute of Physics.
Keywords:Bell Theorem; Geometry; Measurement Theory; Probability
ID Code:42842
Deposited On:07 Jun 2011 04:40
Last Modified:18 May 2016 00:00

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