dorsal/arxiv
View SchemaThe geometry of entanglement witnesses and local detection of entanglement
| Authors | Arthur O. Pittenger, Morton H. Rubin |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0207024 |
| URL | https://arxiv.org/abs/quant-ph/0207024 |
| DOI | 10.1103/PhysRevA.67.012327 |
| Journal | Phys. Rev. A 67, 012327 (2003). |
Abstract
Let $H^{[ N]}=H^{[ d_{1}]}\otimes ... \otimes H^{[ d_{n}]}$ be a tensor product of Hilbert spaces and let $\tau_{0}$ be the closest separable state in the Hilbert-Schmidt norm to an entangled state $\rho_{0}$. Let $\tilde{\tau}_{0}$ denote the closest separable state to $\rho_{0}$ along the line segment from $I/N$ to $\rho_{0}$ where $I$ is the identity matrix. Following [pitrubmat] a witness $W_{0}$ detecting the entanglement of $\rho_{0}$ can be constructed in terms of $I, \tau_{0}$ and $\tilde{\tau}_{0}$. If representations of $\tau_{0}$ and $\tilde{\tau}_{0}$ as convex combinations of separable projections are known, then the entanglement of $\rho_{0}$ can be detected by local measurements. G\"{u}hne \textit{et. al.} in [bruss1] obtain the minimum number of measurement settings required for a class of two qubit states. We use our geometric approach to generalize their result to the corresponding two qudit case when $d$ is prime and obtain the minimum number of measurement settings. In those particular bipartite cases, $\tau_{0}=\tilde{\tau}_{0}$. We illustrate our general approach with a two parameter family of three qubit bound entangled states for which $\tau_{0} \neq \tilde{\tau}_{0}$ and we show our approach works for $n$ qubits. In [pitt] we elaborated on the role of a ``far face'' of the separable states relative to a bound entangled state $\rho_{0}$ constructed from an orthogonal unextendible product base. In this paper the geometric approach leads to an entanglement witness expressible in terms of a constant times $I$ and a separable density $\mu_{0}$ on the far face from $\rho_{0}$. Up to a normalization this coincides with the witness obtained in [bruss1] for the particular example analyzed there.
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"abstract": "Let $H^{[ N]}=H^{[ d_{1}]}\\otimes ... \\otimes H^{[ d_{n}]}$ be a tensor\nproduct of Hilbert spaces and let $\\tau_{0}$ be the closest separable state in\nthe Hilbert-Schmidt norm to an entangled state $\\rho_{0}$. Let\n$\\tilde{\\tau}_{0}$ denote the closest separable state to $\\rho_{0}$ along the\nline segment from $I/N$ to $\\rho_{0}$ where $I$ is the identity matrix.\nFollowing [pitrubmat] a witness $W_{0}$ detecting the entanglement of\n$\\rho_{0}$ can be constructed in terms of $I, \\tau_{0}$ and $\\tilde{\\tau}_{0}$.\nIf representations of $\\tau_{0}$ and $\\tilde{\\tau}_{0}$ as convex combinations\nof separable projections are known, then the entanglement of $\\rho_{0}$ can be\ndetected by local measurements. G\\\"{u}hne \\textit{et. al.} in [bruss1] obtain\nthe minimum number of measurement settings required for a class of two qubit\nstates. We use our geometric approach to generalize their result to the\ncorresponding two qudit case when $d$ is prime and obtain the minimum number of\nmeasurement settings. In those particular bipartite cases,\n$\\tau_{0}=\\tilde{\\tau}_{0}$. We illustrate our general approach with a two\nparameter family of three qubit bound entangled states for which $\\tau_{0} \\neq\n\\tilde{\\tau}_{0}$ and we show our approach works for $n$ qubits.\n In [pitt] we elaborated on the role of a ``far face\u0027\u0027 of the separable states\nrelative to a bound entangled state $\\rho_{0}$ constructed from an orthogonal\nunextendible product base. In this paper the geometric approach leads to an\nentanglement witness expressible in terms of a constant times $I$ and a\nseparable density $\\mu_{0}$ on the far face from $\\rho_{0}$. Up to a\nnormalization this coincides with the witness obtained in [bruss1] for the\nparticular example analyzed there.",
"arxiv_id": "quant-ph/0207024",
"authors": [
"Arthur O. Pittenger",
"Morton H. Rubin"
],
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"quant-ph"
],
"doi": "10.1103/PhysRevA.67.012327",
"journal_ref": "Phys. Rev. A 67, 012327 (2003).",
"title": "The geometry of entanglement witnesses and local detection of entanglement",
"url": "https://arxiv.org/abs/quant-ph/0207024"
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