dorsal/arxiv
View SchemaPhysical properties of the Schur complement of local covariance matrices
| Authors | Luis F. Haruna, Marcos C. de Oliveira |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0701196 |
| URL | https://arxiv.org/abs/quant-ph/0701196 |
| DOI | 10.1088/1751-8113/40/47/011 |
| Journal | J. Phys. A: Math. Theor. 40, 14205 (2007) |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
General properties of global covariance matrices representing bipartite Gaussian states can be decomposed into properties of local covariance matrices and their Schur complements. We demonstrate that given a bipartite Gaussian state $\rho_{12}$ described by a $4\times 4$ covariance matrix \textbf{V}, the Schur complement of a local covariance submatrix $\textbf{V}_1$ of it can be interpreted as a new covariance matrix representing a Gaussian operator of party 1 conditioned to local parity measurements on party 2. The connection with a partial parity measurement over a bipartite quantum state and the determination of the reduced Wigner function is given and an operational process of parity measurement is developed. Generalization of this procedure to a $n$-partite Gaussian state is given and it is demonstrated that the $n-1$ system state conditioned to a partial parity projection is given by a covariance matrix such as its $2 \times 2$ block elements are Schur complements of special local matrices.
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"abstract": "General properties of global covariance matrices representing bipartite\nGaussian states can be decomposed into properties of local covariance matrices\nand their Schur complements. We demonstrate that given a bipartite Gaussian\nstate $\\rho_{12}$ described by a $4\\times 4$ covariance matrix \\textbf{V}, the\nSchur complement of a local covariance submatrix $\\textbf{V}_1$ of it can be\ninterpreted as a new covariance matrix representing a Gaussian operator of\nparty 1 conditioned to local parity measurements on party 2. The connection\nwith a partial parity measurement over a bipartite quantum state and the\ndetermination of the reduced Wigner function is given and an operational\nprocess of parity measurement is developed. Generalization of this procedure to\na $n$-partite Gaussian state is given and it is demonstrated that the $n-1$\nsystem state conditioned to a partial parity projection is given by a\ncovariance matrix such as its $2 \\times 2$ block elements are Schur complements\nof special local matrices.",
"arxiv_id": "quant-ph/0701196",
"authors": [
"Luis F. Haruna",
"Marcos C. de Oliveira"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"doi": "10.1088/1751-8113/40/47/011",
"journal_ref": "J. Phys. A: Math. Theor. 40, 14205 (2007)",
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Physical properties of the Schur complement of local covariance matrices",
"url": "https://arxiv.org/abs/quant-ph/0701196"
},
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