Physical properties of the Schur complement of local covariance matrices

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 ${\text{V}}_1$ 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.

arXiv comment10 pages. Replaced with final published version

References in corpus (16)

• A computable measure of entanglement
• Peres-Horodecki separability criterion for continuous variable systems
• On the impossibility of distilling Gaussian states with Gaussian operations
• The characterization of Gaussian operations and Distillation of Gaussian States
• Gaussian transformations and distillation of entangled Gaussian states
• Direct Probing of Quantum Phase Space by Photon Counting
• Entanglement of formation for symmetric Gaussian states
• Direct measurement of the Wigner function by photon counting
• Quantum computation with mesoscopic superposition states
• Gaussian quantum channels
• On the P-representable subset of all bipartite Gaussian separable states
• Lower bounds on the entanglement of formation for general Gaussian states
• Pure State Correlations: Chords in Phase Space
• Characterization and quantification of symmetric Gaussian state entanglement through a local classicality criterion
• Discrete teleportation protocol of continuum spectra field states
• Parity Measurements, Decoherence and Spiky Wigner Functions

Cited by in corpus (2)

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