dorsal/arxiv
View SchemaBell Inequalities in Four Dimensional Phase Space and the Three Marginal Theorem
| Authors | G. Auberson, ; G. Mahoux, ; S. M. Roy, Virendra Singh |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0205185 |
| URL | https://arxiv.org/abs/quant-ph/0205185 |
| DOI | 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 qi and momentum variables pj, 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 rho({qi},{pj}) reproducing them as marginals; this settles a long standing conjecture; it is achieved by first deriving Bell-like 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 rho({qi},{pj}); the solution is not unique and its general form is worked out. These two points constitute our ``three marginal theorem''.
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"abstract": "We address the classical and quantum marginal problems, namely the question\nof simultaneous realizability through a common probability density in phase\nspace of a given set of compatible probability distributions. We consider only\ndistributions authorized by quantum mechanics, i.e. those corresponding to\ncomplete commuting sets of observables. For four-dimensional phase space with\nposition variables qi and momentum variables pj, we establish the two following\npoints: i) given four compatible probabilities for (q1,q2), (q1,p2), (p1,q2)\nand (p1,p2), there does not always exist a positive phase space density\nrho({qi},{pj}) reproducing them as marginals; this settles a long standing\nconjecture; it is achieved by first deriving Bell-like inequalities in phase\nspace which have their own theoretical and experimental interest. ii) given\ninstead at most three compatible probabilities, there always exist an\nassociated phase space density rho({qi},{pj}); the solution is not unique and\nits general form is worked out. These two points constitute our ``three\nmarginal theorem\u0027\u0027.",
"arxiv_id": "quant-ph/0205185",
"authors": [
"G. Auberson",
"; G. Mahoux",
"; S. M. Roy",
"Virendra Singh"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.1578532",
"title": "Bell Inequalities in Four Dimensional Phase Space and the Three Marginal Theorem",
"url": "https://arxiv.org/abs/quant-ph/0205185"
},
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