dorsal/arxiv
View SchemaSeparability and Fourier representations of density matrices
| Authors | Arthur O. Pittenger, Morton H. Rubin |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0001014 |
| URL | https://arxiv.org/abs/quant-ph/0001014 |
| DOI | 10.1103/PhysRevA.62.032313 |
| Journal | Phys. Rev. A62, 032313 (2000). |
Abstract
Using the finite Fourier transform, we introduce a generalization of Pauli-spin matrices for $d$-dimensional spaces, and the resulting set of unitary matrices $S(d) $ is a basis for $d\times d$ matrices. If $N=d_{1}\times d_{2}\times...\times d_{b}$ and $H^{[ N]}=\bigotimes H^{% [ d_{k}]}$, we give a sufficient condition for separability of a density matrix $\rho $ relative to the $H^{[ d_{k}]}$ in terms of the $L_{1}$ norm of the spin coefficients of $\rho >.$ Since the spin representation depends on the form of the tensor product, the theory applies to both full and partial separability on a given space $H^{[ N]}$% . It follows from this result that for a prescribed form of separability, there is always a neighborhood of the normalized identity in which every density matrix is separable. We also show that for every prime $p$ and $n>1$ the generalized Werner density matrix $W^{[ p^{n}]}(s) $ is fully separable if and only if $s\leq (1+p^{n-1}) ^{-1}$.
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"abstract": "Using the finite Fourier transform, we introduce a generalization of\nPauli-spin matrices for $d$-dimensional spaces, and the resulting set of\nunitary matrices $S(d) $ is a basis for $d\\times d$ matrices. If $N=d_{1}\\times\nd_{2}\\times...\\times d_{b}$ and $H^{[ N]}=\\bigotimes H^{% [ d_{k}]}$, we give a\nsufficient condition for separability of a density matrix $\\rho $ relative to\nthe $H^{[ d_{k}]}$ in terms of the $L_{1}$ norm of the spin coefficients of\n$\\rho \u003e.$ Since the spin representation depends on the form of the tensor\nproduct, the theory applies to both full and partial separability on a given\nspace $H^{[ N]}$% . It follows from this result that for a prescribed form of\nseparability, there is always a neighborhood of the normalized identity in\nwhich every density matrix is separable. We also show that for every prime $p$\nand $n\u003e1$ the generalized Werner density matrix $W^{[ p^{n}]}(s) $ is fully\nseparable if and only if $s\\leq (1+p^{n-1}) ^{-1}$.",
"arxiv_id": "quant-ph/0001014",
"authors": [
"Arthur O. Pittenger",
"Morton H. Rubin"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.62.032313",
"journal_ref": "Phys. Rev. A62, 032313 (2000).",
"title": "Separability and Fourier representations of density matrices",
"url": "https://arxiv.org/abs/quant-ph/0001014"
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