dorsal/arxiv
View SchemaOn the degree conjecture for separability of multipartite quantum states
| Authors | Ali Saif M. Hassan, Pramod Joag |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0701040 |
| URL | https://arxiv.org/abs/quant-ph/0701040 |
| DOI | 10.1063/1.2830978 |
| Journal | JOURNAL OF MATHEMATICAL PHYSICS 49, 012105 (2008) |
Abstract
We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein {\it et al.} [Phys. Rev. A \textbf{73}, 012320 (2006)]. The conjecture states that a multipartite quantum state is separable if and only if the degree matrix of the graph associated with the state is equal to the degree matrix of the partial transpose of this graph. We call this statement to be the strong form of the conjecture. In its weak version, the conjecture requires only the necessity, that is, if the state is separable, the corresponding degree matrices match. We prove the strong form of the conjecture for {\it pure} multipartite quantum states, using the modified tensor product of graphs defined in [J. Phys. A: Math. Theor. \textbf{40}, 10251 (2007)], as both necessary and sufficient condition for separability. Based on this proof, we give a polynomial-time algorithm for completely factorizing any pure multipartite quantum state. By polynomial-time algorithm we mean that the execution time of this algorithm increases as a polynomial in $m,$ where $m$ is the number of parts of the quantum system. We give a counter-example to show that the conjecture fails, in general, even in its weak form, for multipartite mixed states. Finally, we prove this conjecture, in its weak form, for a class of multipartite mixed states, giving only a necessary condition for separability.
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"abstract": "We settle the so-called degree conjecture for the separability of\nmultipartite quantum states, which are normalized graph Laplacians, first given\nby Braunstein {\\it et al.} [Phys. Rev. A \\textbf{73}, 012320 (2006)]. The\nconjecture states that a multipartite quantum state is separable if and only if\nthe degree matrix of the graph associated with the state is equal to the degree\nmatrix of the partial transpose of this graph. We call this statement to be the\nstrong form of the conjecture. In its weak version, the conjecture requires\nonly the necessity, that is, if the state is separable, the corresponding\ndegree matrices match. We prove the strong form of the conjecture for {\\it\npure} multipartite quantum states, using the modified tensor product of graphs\ndefined in [J. Phys. A: Math. Theor. \\textbf{40}, 10251 (2007)], as both\nnecessary and sufficient condition for separability. Based on this proof, we\ngive a polynomial-time algorithm for completely factorizing any pure\nmultipartite quantum state. By polynomial-time algorithm we mean that the\nexecution time of this algorithm increases as a polynomial in $m,$ where $m$ is\nthe number of parts of the quantum system. We give a counter-example to show\nthat the conjecture fails, in general, even in its weak form, for multipartite\nmixed states. Finally, we prove this conjecture, in its weak form, for a class\nof multipartite mixed states, giving only a necessary condition for\nseparability.",
"arxiv_id": "quant-ph/0701040",
"authors": [
"Ali Saif M. Hassan",
"Pramod Joag"
],
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"quant-ph"
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"doi": "10.1063/1.2830978",
"journal_ref": "JOURNAL OF MATHEMATICAL PHYSICS 49, 012105 (2008)",
"title": "On the degree conjecture for separability of multipartite quantum states",
"url": "https://arxiv.org/abs/quant-ph/0701040"
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