dorsal/arxiv
View SchemaQuantum conditional operator and a criterion for separability
| Authors | N. J. Cerf, C. Adami, R. M. Gingrich |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9710001 |
| URL | https://arxiv.org/abs/quant-ph/9710001 |
| DOI | 10.1103/PhysRevA.60.893 |
| Journal | Phys.Rev.A60:893-898,1999 |
Abstract
We analyze the properties of the conditional amplitude operator, the quantum analog of the conditional probability which has been introduced in [quant-ph/9512022]. The spectrum of the conditional operator characterizing a quantum bipartite system is invariant under local unitary transformations and reflects its inseparability. More specifically, it is shown that the conditional amplitude operator of a separable state cannot have an eigenvalue exceeding 1, which results in a necessary condition for separability. This leads us to consider a related separability criterion based on the positive map $\Gamma:\rho \to (Tr \rho) - \rho$, where $\rho$ is an Hermitian operator. Any separable state is mapped by the tensor product of this map and the identity into a non-negative operator, which provides a simple necessary condition for separability. In the special case where one subsystem is a quantum bit, $\Gamma$ reduces to time-reversal, so that this separability condition is equivalent to partial transposition. It is therefore also sufficient for $2\times 2$ and $2\times 3$ systems. Finally, a simple connection between this map and complex conjugation in the "magic" basis is displayed.
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"abstract": "We analyze the properties of the conditional amplitude operator, the quantum\nanalog of the conditional probability which has been introduced in\n[quant-ph/9512022]. The spectrum of the conditional operator characterizing a\nquantum bipartite system is invariant under local unitary transformations and\nreflects its inseparability. More specifically, it is shown that the\nconditional amplitude operator of a separable state cannot have an eigenvalue\nexceeding 1, which results in a necessary condition for separability. This\nleads us to consider a related separability criterion based on the positive map\n$\\Gamma:\\rho \\to (Tr \\rho) - \\rho$, where $\\rho$ is an Hermitian operator. Any\nseparable state is mapped by the tensor product of this map and the identity\ninto a non-negative operator, which provides a simple necessary condition for\nseparability. In the special case where one subsystem is a quantum bit,\n$\\Gamma$ reduces to time-reversal, so that this separability condition is\nequivalent to partial transposition. It is therefore also sufficient for\n$2\\times 2$ and $2\\times 3$ systems. Finally, a simple connection between this\nmap and complex conjugation in the \"magic\" basis is displayed.",
"arxiv_id": "quant-ph/9710001",
"authors": [
"N. J. Cerf",
"C. Adami",
"R. M. Gingrich"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.60.893",
"journal_ref": "Phys.Rev.A60:893-898,1999",
"title": "Quantum conditional operator and a criterion for separability",
"url": "https://arxiv.org/abs/quant-ph/9710001"
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