dorsal/arxiv
View SchemaConcurrence of Lorentz-positive maps
| Authors | Roland Hildebrand |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0612064 |
| URL | https://arxiv.org/abs/quant-ph/0612064 |
Abstract
Let L_n be the n-dimensional Lorentz cone. A linear map M from R^m to R^n is called Lorentz-positive if M[L_m] is contained in L_n. We extend the notion of concurrence, which was initially introduced to quantify the entanglement of bipartite density matrices, to Lorentz-positive maps and provide an explicite formula for it. This allows us to obtain formulae for the concurrence of arbitrary positive operators taking 2 x 2 complex hermitian matrices as input and consequently of arbitrary bipartite density matrices of rank 2. Namely, let P: H(2) \to H(d) be a positive operator, and let \lambda_1,...,\lambda_4 be the generalized eigenvalues of the pencil \sigma_2(P(X)) - \lambda det X, in decreasing order, where \sigma_2 is the second symmetric function of the spectrum. Then the concurrence is given by the expression C(P;X) = 2\sqrt{\sigma_2(P(X)) - \lambda_2 det X}. As an application, we compute the concurrences of the density matrices of all graphs with 2 edges. Similar results apply for a function which we call I-fidelity, with the second largest generalized eigenvalue \lambda_2 replaced by the smallest eigenvalue \lambda_4.
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"abstract": "Let L_n be the n-dimensional Lorentz cone. A linear map M from R^m to R^n is\ncalled Lorentz-positive if M[L_m] is contained in L_n. We extend the notion of\nconcurrence, which was initially introduced to quantify the entanglement of\nbipartite density matrices, to Lorentz-positive maps and provide an explicite\nformula for it. This allows us to obtain formulae for the concurrence of\narbitrary positive operators taking 2 x 2 complex hermitian matrices as input\nand consequently of arbitrary bipartite density matrices of rank 2. Namely, let\nP: H(2) \\to H(d) be a positive operator, and let \\lambda_1,...,\\lambda_4 be the\ngeneralized eigenvalues of the pencil \\sigma_2(P(X)) - \\lambda det X, in\ndecreasing order, where \\sigma_2 is the second symmetric function of the\nspectrum. Then the concurrence is given by the expression C(P;X) =\n2\\sqrt{\\sigma_2(P(X)) - \\lambda_2 det X}. As an application, we compute the\nconcurrences of the density matrices of all graphs with 2 edges. Similar\nresults apply for a function which we call I-fidelity, with the second largest\ngeneralized eigenvalue \\lambda_2 replaced by the smallest eigenvalue \\lambda_4.",
"arxiv_id": "quant-ph/0612064",
"authors": [
"Roland Hildebrand"
],
"categories": [
"quant-ph"
],
"title": "Concurrence of Lorentz-positive maps",
"url": "https://arxiv.org/abs/quant-ph/0612064"
},
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