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

Fortschritte der Physik

Entanglement properties of Gaussian states

Barbara Kraus, G. Giedke, Maciej Lewenstein, J. I. Cirac

Published Apr 24, 2003• 3 citations
Publisher

Abstract

Abstract On the one hand we present necessary and sufficient conditions for both the separability and the distillability of bipartite Gaussian states of an arbitrary number of modes. On the other hand we classify tripartite three‐mode Gaussian states according to their separability properties and provide necessary and sufficient conditions for separability of those states . All the conditions presented here can be easily checked by direct calculations, thus providing operational criteria. This solves the separability problem for all bipartite as well as tripartite three‐mode Gaussian states and the distillability problem for all bipartite Gaussian states.

DOI
10.1002/prop.200310043

Related papers

Separability and Distillability of bipartite Gaussian States – the Complete Story

G. Giedke, Barbara Kraus, L.-M. Duan +3 more

We present necessary and sufficient conditions for both the separability and the distillability of bipartite Gaussian states of an arbitrary number of modes. The conditions can be easily checked by direct calculations, thus providing operational criteria for both properties. This solves the separability and distillability problems for Gaussian states.

Fortschritte der Physik•Oct 1, 2001•15 citations

Entanglement Criteria for All Bipartite Gaussian States

G. Giedke, Barbara Kraus, Maciej Lewenstein +1 more

We provide a necessary and sufficient condition for separability of Gaussian states of bipartite systems of arbitrarily many modes. The condition provides an operational criterion since it can be checked by simple computation. Moreover, it allows us to find a pure product-state decomposition of any given separable Gaussian state. We also show that all bipartite Gaussian states with nonpositive partial transpose are distillable.

arXiv:quant-ph/0104050•Physical Review Letters•Oct 1, 2001•190 citations

Separability properties of three-mode Gaussian states

G. Giedke, Barbara Kraus, Maciej Lewenstein +1 more

We derive a necessary and sufficient condition for the separability of tripartite three-mode Gaussian states that is easy to check for any such state. We give a classification of the separability properties of those systems and show how to determine for any state to which class it belongs. We show that there exist genuinely tripartite bound entangled states and point out how to construct and prepare such states.

arXiv:quant-ph/0103137•Physical Review A•Oct 8, 2001•195 citations

Tripartite separability conditions exponentially violated by Gaussian states

Evgeny V Shchukin, Peter van Loock

Starting with a set of conditions for bipartite separability of arbitrary quantum states in any dimension and expressed in terms of arbitrary operators whose commutator is a $c$-number, we derive a hierarchy of conditions for tripartite separability of continuous-variable three-mode quantum states. These conditions have the form of inequalities for higher-order moments of linear combinations of the mode operators. They enable one to distinguish between all possible kinds of tripartite separability, while the strongest violation of these inequalities is a sufficient condition for genuine tripartite entanglement. We construct Gaussian states for which the violation of our conditions grows exponentially with the order of the moments of the mode operators. By going beyond second moments, our conditions are expected to be useful as well for the detection of tripartite entanglement of non-Gaussian states. We also demonstrate that our conditions can be easily implemented experimentally.

arXiv:1402.1701•Physical Review A•Jul 28, 2014•5 citations

Bound Entangled Gaussian States

Reinhard F. Werner, Michael M. Wolf

We discuss the entanglement properties of bipartite states with Gaussian Wigner functions. For the separability, and the positivity of the partial transpose, we establish explicit necessary and sufficient criteria in terms of the covariance matrix of the state. It is shown that, for systems composed of a single oscillator for Alice and an arbitrary number for Bob, positivity of the partial transpose implies separability. However, this implication fails with two oscillators on each side, as we show by constructing a five parameter family of bound entangled Gaussian states.

arXiv:quant-ph/0009118•Physical Review Letters•Apr 16, 2001•459 citations

Entanglement in continuous-variable systems: recent advances and current perspectives

Gerardo Adesso, Fabrizio Illuminati

We review the theory of continuous-variable entanglement with special emphasis on foundational aspects, conceptual structures and mathematical methods. Much attention is devoted to the discussion of separability criteria and entanglement properties of Gaussian states, for their great practical relevance in applications to quantum optics and quantum information, as well as for the very clean framework that they allow for the study of the structure of nonlocal correlations. We give a self-contained introduction to phase-space and symplectic methods in the study of Gaussian states of infinite-dimensional bosonic systems. We review the most important results on the separability and distillability of Gaussian states and discuss the main properties of bipartite entanglement. These include the extremal entanglement, minimal and maximal, of two-mode mixed Gaussian states, the ordering of two-mode Gaussian states according to different measures of entanglement, the unitary (reversible) localization and the scaling of bipartite entanglement in multimode Gaussian states. We then discuss recent advances in the understanding of entanglement sharing in multimode Gaussian states, including the proof of the monogamy inequality of distributed entanglement for all Gaussian states. Multipartite entanglement of Gaussian states is reviewed by discussing its qualification by different classes of separability, and the main consequences of the monogamy inequality, such as the quantification of genuine tripartite entanglement in three-mode Gaussian states, the promiscuous nature of entanglement sharing in symmetric Gaussian states and the possible coexistence of unlimited bipartite and multipartite entanglement. We finally review recent advances and discuss possible perspectives on the qualification and quantification of entanglement in non-Gaussian states, a field of research that is to a large extent yet to be explored.

arXiv:quant-ph/0701221•Journal of Physics A Mathematical and Theoretical•Jun 27, 2007•744 citations
Abstract On the one hand we present necessary and sufficient conditions for both the separability and the distillability of bipartite Gaussian states of an arbitrary number of modes. On the other hand we classify tripartite three‐mode Gaussian states according to their separability properties and provide necessary and sufficient conditions for separability of those states . All the conditions presented here can be easily checked by direct calculations, thus providing operational criteria. This solves the separability problem for all bipartite as well as tripartite three‐mode Gaussian states and the distillability problem for all bipartite Gaussian states.
Starting with a set of conditions for bipartite separability of arbitrary quantum states in any dimension and expressed in terms of arbitrary operators whose commutator is a ccc-number, we derive a hierarchy of conditions for tripartite separability of continuous-variable three-mode quantum states. These conditions have the form of inequalities for higher-order moments of linear combinations of the mode operators. They enable one to distinguish between all possible kinds of tripartite separability, while the strongest violation of these inequalities is a sufficient condition for genuine tripartite entanglement. We construct Gaussian states for which the violation of our conditions grows exponentially with the order of the moments of the mode operators. By going beyond second moments, our conditions are expected to be useful as well for the detection of tripartite entanglement of non-Gaussian states. We also demonstrate that our conditions can be easily implemented experimentally.
We discuss the entanglement properties of bipartite states with Gaussian Wigner functions. For the separability, and the positivity of the partial transpose, we establish explicit necessary and sufficient criteria in terms of the covariance matrix of the state. It is shown that, for systems composed of a single oscillator for Alice and an arbitrary number for Bob, positivity of the partial transpose implies separability. However, this implication fails with two oscillators on each side, as we show by constructing a five parameter family of bound entangled Gaussian states.
We review the theory of continuous-variable entanglement with special emphasis on foundational aspects, conceptual structures and mathematical methods. Much attention is devoted to the discussion of separability criteria and entanglement properties of Gaussian states, for their great practical relevance in applications to quantum optics and quantum information, as well as for the very clean framework that they allow for the study of the structure of nonlocal correlations. We give a self-contained introduction to phase-space and symplectic methods in the study of Gaussian states of infinite-dimensional bosonic systems. We review the most important results on the separability and distillability of Gaussian states and discuss the main properties of bipartite entanglement. These include the extremal entanglement, minimal and maximal, of two-mode mixed Gaussian states, the ordering of two-mode Gaussian states according to different measures of entanglement, the unitary (reversible) localization and the scaling of bipartite entanglement in multimode Gaussian states. We then discuss recent advances in the understanding of entanglement sharing in multimode Gaussian states, including the proof of the monogamy inequality of distributed entanglement for all Gaussian states. Multipartite entanglement of Gaussian states is reviewed by discussing its qualification by different classes of separability, and the main consequences of the monogamy inequality, such as the quantification of genuine tripartite entanglement in three-mode Gaussian states, the promiscuous nature of entanglement sharing in symmetric Gaussian states and the possible coexistence of unlimited bipartite and multipartite entanglement. We finally review recent advances and discuss possible perspectives on the qualification and quantification of entanglement in non-Gaussian states, a field of research that is to a large extent yet to be explored.