dorsal/arxiv
View SchemaMarginal distributions in $(\bf 2N)$-dimensional phase space and the quantum $(\bf N+1)$ marginal theorem
| Authors | G. Auberson, G. Mahoux, S. M. Roy, V. Singh |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0402113 |
| URL | https://arxiv.org/abs/quant-ph/0402113 |
| DOI | 10.1063/1.1807954 |
Abstract
We study the problem of constructing a probability density in 2N-dimensional phase space which reproduces a given collection of $n$ joint probability distributions as marginals. Only distributions authorized by quantum mechanics, i.e. depending on a (complete) commuting set of $N$ variables, are considered. A diagrammatic or graph theoretic formulation of the problem is developed. We then exactly determine the set of ``admissible'' data, i.e. those types of data for which the problem always admits solutions. This is done in the case where the joint distributions originate from quantum mechanics as well as in the case where this constraint is not imposed. In particular, it is shown that a necessary (but not sufficient) condition for the existence of solutions is $n\leq N+1$. When the data are admissible and the quantum constraint is not imposed, the general solution for the phase space density is determined explicitly. For admissible data of a quantum origin, the general solution is given in certain (but not all) cases. In the remaining cases, only a subset of solutions is obtained.
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"abstract": "We study the problem of constructing a probability density in 2N-dimensional\nphase space which reproduces a given collection of $n$ joint probability\ndistributions as marginals. Only distributions authorized by quantum mechanics,\ni.e. depending on a (complete) commuting set of $N$ variables, are considered.\nA diagrammatic or graph theoretic formulation of the problem is developed. We\nthen exactly determine the set of ``admissible\u0027\u0027 data, i.e. those types of data\nfor which the problem always admits solutions. This is done in the case where\nthe joint distributions originate from quantum mechanics as well as in the case\nwhere this constraint is not imposed. In particular, it is shown that a\nnecessary (but not sufficient) condition for the existence of solutions is\n$n\\leq N+1$. When the data are admissible and the quantum constraint is not\nimposed, the general solution for the phase space density is determined\nexplicitly. For admissible data of a quantum origin, the general solution is\ngiven in certain (but not all) cases. In the remaining cases, only a subset of\nsolutions is obtained.",
"arxiv_id": "quant-ph/0402113",
"authors": [
"G. Auberson",
"G. Mahoux",
"S. M. Roy",
"V. Singh"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.1807954",
"title": "Marginal distributions in $(\\bf 2N)$-dimensional phase space and the quantum $(\\bf N+1)$ marginal theorem",
"url": "https://arxiv.org/abs/quant-ph/0402113"
},
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