Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2003.11049

A Note on the Disentanglement of Gaussian Quantum States by Symplectic Rotations

Maurice A. de Gosson

Published Mar 24, 2020• 0 citations
arXivHTMLPDF

Abstract

We show that every Gaussian mixed quantum state can be disentangled by conjugation with a metaplectic operator associated with a symplectic rotation. The main tools we use are the Werner-Wolf condition on covariance matrices and the symplectic covariance of Weyl quantization.
DOI
10.48550/arxiv.2003.11049

Related papers

On the disentanglement of Gaussian quantum states by symplectic rotations

Maurice A. de Gosson

We show that every Gaussian mixed quantum state can be disentangled by conjugation with a unitary operator corresponding to a symplectic rotation via the metaplectic representation of the symplectic group. The main tools we use are the Werner–Wolf condition for separability on covariance matrices and the symplectic covariance of Weyl pseudo-differential operators.

Comptes Rendus Mathématique•Jul 28, 2020•4 citations

Gaussian quantum states can be disentangled using symplectic rotations

Maurice A. de Gosson

We show that every Gaussian mixed quantum state can be disentangled by conjugation with a passive symplectic transformation, that is a metaplectic operator associated with a symplectic rotation. The main tools we use are the Werner-Wolf condition on covariance matrices and the symplectic covariance of Weyl quantization. Our result therefore complements a recent study by Lami, Serafini, and Adesso.

Letters in Mathematical Physics•May 24, 2021•3 citations

Entanglement of Bipartite Gaussian States: a Simple Criterion and its Geometric Interpretation

Maurice A. de Gosson

Werner and Wolf have proven in Phys. Rev. Lett. 86(16) (2001) a very elegant necessary and sufficient condition for a bosonic continuous variable bipartite Gaussian mixed quantum state to be separable. This condition is, however, difficult to implement in practice. In the present Letter, we propose a simpler condition which only involves the calculation of the symplectic matrix in the Williamson diagonalization of the covariance matrix of the state under consideration. The main tool in our construction is the observation, proved in previous work, that the Wigner transform is covariant only under symplectic or antisymplectic linear transformations. We also give a geometric interpretation of our condition in terms of the orthogonal projections of "quantum blobs"..

arXiv:2210.01735•Oct 4, 2022•0 citations

A Symplectic Interpretation of the Separability of Gaussian Mixed States

Maurice A. de Gosson

We prove, using symplectic methods and The Wigner formalism, a refinement of a criterion due to Werner and Wolf for the separability of bipartite Gaussian mixed states in an arbitrary number of dimensions. We use our result to show that one can characterize separability by comparing these states with separable pure Gaussian states.

arXiv:1809.00184•Sep 1, 2018•0 citations

On the covariance matrix for Gaussian states

Angel García-Chung

We show the explicit expression for the covariance matrix of general Gaussian states in terms of the symplectic group matrices. We discuss how the criteria to characterize squeezing and entanglement using the covariance matrix give rise to new criteria in the symplectic matrix elements used to construct the general Gaussian states.

arXiv:2003.11063•Mar 24, 2020•0 citations

A result about the classification of quantum covariance matrices based on their eigenspectra

Arik Avagyan

The set of covariance matrices of a continuous-variable quantum system with a finite number of degrees of freedom is a strict subset of the set of real positive-definite matrices (PDMs) due to Heisenberg’s uncertainty principle. This has the implication that, in general, not every orthogonal transform of a quantum covariance matrix (CM) produces a PDM that obeys the uncertainty principle. A natural question thus arises, to find the set of quantum covariance matrices consistent with a given eigenspectrum. For the special class of pure Gaussian states the set of quantum covariance matrices with a given eigenspectrum consists of a single orbit of the action of the orthogonal symplectic group. The eigenspectrum of a CM of a state in this class is composed of pairs that each multiply to one. Our main contribution is finding a non-trivial class of eigenspectra with the property that the set of quantum covariance matrices corresponding to any eigenspectrum in this class are related by orthogonal symplectic transformations. We show that all non-degenerate eigenspectra with this property must belong to this class, and that the set of such eigenspectra coincides with the class of non-degenerate eigenspectra that identify the physically relevant thermal and squeezing parameters of a Gaussian state.

arXiv:2308.03439•Reviews in Mathematical Physics•Jan 27, 2024•1 citation
Werner and Wolf have proven in Phys. Rev. Lett. 86(16) (2001) a very elegant necessary and sufficient condition for a bosonic continuous variable bipartite Gaussian mixed quantum state to be separable. This condition is, however, difficult to implement in practice. In the present Letter, we propose a simpler condition which only involves the calculation of the symplectic matrix in the Williamson diagonalization of the covariance matrix of the state under consideration. The main tool in our construction is the observation, proved in previous work, that the Wigner transform is covariant only under symplectic or antisymplectic linear transformations. We also give a geometric interpretation of our condition in terms of the orthogonal projections of "quantum blobs"..
The set of covariance matrices of a continuous-variable quantum system with a finite number of degrees of freedom is a strict subset of the set of real positive-definite matrices (PDMs) due to Heisenberg’s uncertainty principle. This has the implication that, in general, not every orthogonal transform of a quantum covariance matrix (CM) produces a PDM that obeys the uncertainty principle. A natural question thus arises, to find the set of quantum covariance matrices consistent with a given eigenspectrum. For the special class of pure Gaussian states the set of quantum covariance matrices with a given eigenspectrum consists of a single orbit of the action of the orthogonal symplectic group. The eigenspectrum of a CM of a state in this class is composed of pairs that each multiply to one. Our main contribution is finding a non-trivial class of eigenspectra with the property that the set of quantum covariance matrices corresponding to any eigenspectrum in this class are related by orthogonal symplectic transformations. We show that all non-degenerate eigenspectra with this property must belong to this class, and that the set of such eigenspectra coincides with the class of non-degenerate eigenspectra that identify the physically relevant thermal and squeezing parameters of a Gaussian state.