dorsal/arxiv
View SchemaOperational criterion and constructive checks for the separability of low rank density matrices
| Authors | Pawel Horodecki, Maciej Lewenstein, Guifré Vidal, Ignacio Cirac |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0002089 |
| URL | https://arxiv.org/abs/quant-ph/0002089 |
| DOI | 10.1103/PhysRevA.62.032310 |
| Journal | Phys. Rev. A 62, 032310 (2000) |
Abstract
We consider low rank density operators $\varrho$ supported on a $M\times N$ Hilbert space for arbitrary $M$ and $N$ ($M\leq N$) and with a positive partial transpose (PPT) $\varrho^{T_A}\ge 0$. For rank $r(\varrho) \leq N$ we prove that having a PPT is necessary and sufficient for $\varrho$ to be separable; in this case we also provide its minimal decomposition in terms of pure product states. It follows from this result that there is no rank 3 bound entangled states having a PPT. We also present a necessary and sufficient condition for the separability of generic density matrices for which the sum of the ranks of $\varrho$ and $\varrho^{T_A}$ satisfies $r(\varrho)+r(\varrho^{T_A}) \le 2MN-M-N+2$. This separability condition has the form of a constructive check, providing thus also a pure product state decomposition for separable states, and it works in those cases where a system of couple polynomial equations has a finite number of solutions, as expected in most cases.
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"abstract": "We consider low rank density operators $\\varrho$ supported on a $M\\times N$\nHilbert space for arbitrary $M$ and $N$ ($M\\leq N$) and with a positive partial\ntranspose (PPT) $\\varrho^{T_A}\\ge 0$. For rank $r(\\varrho) \\leq N$ we prove\nthat having a PPT is necessary and sufficient for $\\varrho$ to be separable; in\nthis case we also provide its minimal decomposition in terms of pure product\nstates. It follows from this result that there is no rank 3 bound entangled\nstates having a PPT. We also present a necessary and sufficient condition for\nthe separability of generic density matrices for which the sum of the ranks of\n$\\varrho$ and $\\varrho^{T_A}$ satisfies $r(\\varrho)+r(\\varrho^{T_A}) \\le\n2MN-M-N+2$. This separability condition has the form of a constructive check,\nproviding thus also a pure product state decomposition for separable states,\nand it works in those cases where a system of couple polynomial equations has a\nfinite number of solutions, as expected in most cases.",
"arxiv_id": "quant-ph/0002089",
"authors": [
"Pawel Horodecki",
"Maciej Lewenstein",
"Guifr\u00e9 Vidal",
"Ignacio Cirac"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.62.032310",
"journal_ref": "Phys. Rev. A 62, 032310 (2000)",
"title": "Operational criterion and constructive checks for the separability of low rank density matrices",
"url": "https://arxiv.org/abs/quant-ph/0002089"
},
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