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

AIP conference proceedings

Covariance matrix criterion for separability

Oleg Gittsovich, Otfried Gühne, P. Hyllus, Jens Eisert, Alexander I. Lvovsky

Published Jan 1, 2009• 3 citations
Publisher

Abstract

We review an approach to the separability problem using covariance matrices of locally measurable observables. This approach leads to a strong entanglement criterion which we call covariance matrix criterion (CMC). We consider some corollaries of the CMC that in general allow to detect many bound entangled states. In particular we give a new example which sheds some light on the connections between different entanglement criteria derived from the general CMC.

DOI
10.1063/1.3131377

Related papers

Unifying several separability conditions using the covariance matrix criterion

Oleg Gittsovich, Otfried Gühne, P. Hyllus +1 more

We present a framework for deciding whether a quantum state is separable or entangled using covariance matrices of locally measurable observables. This leads to the covariance matrix criterion as a general separability criterion. We demonstrate that this criterion allows us to detect many states where the familiar criterion of the positivity of the partial transpose fails. It turns out that a large number of criteria that have been proposed to complement the positive partial transpose criterion---the computable cross norm or realignment criterion, the criterion based on local uncertainty relations, criteria derived from extensions of the realignment map, and others---are in fact a corollary of the covariance matrix criterion.

arXiv:0803.0757•Physical Review A•Nov 12, 2008•101 citations

Covariance Matrices and the Separability Problem

Otfried Gühne, P. Hyllus, Oleg Gittsovich +1 more

We propose a unifying approach to the separability problem using covariance matrices of locally measurable observables. From a practical point of view, our approach leads to strong entanglement criteria that allow us to detect the entanglement of many bound entangled states in higher dimensions and which are at the same time necessary and sufficient for two qubits. From a fundamental perspective, our approach leads to insights into the relations between several known entanglement criteria--such as the computable cross-norm and local uncertainty criteria--as well as their limitations.

arXiv:quant-ph/0611282•Physical Review Letters•Sep 28, 2007•168 citations

Multiparticle covariance matrices and the impossibility of detecting graph-state entanglement with two-particle correlations

Oleg Gittsovich, Philipp Hyllus, Otfried Gühne

We present a criterion for multiparticle entanglement based on covariance matrices. On the one hand, the criterion allows detection of bound entangled states which are not detected by other criteria; on the other hand, some strongly entangled pure states such as the Greenberger-Horne-Zeilinger (GHZ) states are not detected. We show, however, that this is a general phenomenon: No separability criterion based on two-particle correlations can recognize the entanglement in the family of graph states to which the GHZ states belong.

arXiv:1006.1594•Physical Review A•Sep 9, 2010•53 citations

Entanglement Detection: Complexity and Shannon Entropic Criteria

Yichen Huang

Entanglement plays a key role in quantum information. The separability problem arises: how to prove that a given quantum state is entangled? Covariance-matrix-based entanglement criteria are the most important and widely used approach to this problem in continuous variable systems. Here, the separability problem is systematically studied in the more experimentally realistic setting that the questioned state is only partly known. A simple general formalism is proposed for this variant of the separability problem. It builds relations between certain classes of criteria and bounds the power of these criteria. In particular, it shows the equivalence of many seemingly different covariance-matrix-based criteria; it offers guidance on the study of and reveals the inherent limitations of covariance-matrix-based criteria. Also, a Shannon entropic criterion is proposed for m ⊕ n-mode states. It unifies and improves many known covariance-matrix-based criteria and Shannon entropic criteria; it is necessary and sufficient for 1 ⊕ n-mode Gaussian states and m ⊕ n-mode bisymmetric Gaussian states.

IEEE Transactions on Information Theory•May 21, 2013•19 citations

Bounding entanglement dimensionality from the covariance matrix

Shuheng Liu, Matteo Fadel, Qiongyi He +2 more

High-dimensional entanglement has been identified as an important resource in quantum information processing, and also as a main obstacle for simulating quantum systems. Its certification is often difficult, and most widely used methods for experiments are based on fidelity measurements with respect to highly entangled states. Here, instead, we consider covariances of collective observables, as in the well-known Covariance Matrix Criterion (CMC)[1] and present a generalization of the CMC for determining the Schmidt number of a bipartite system. This is potentially particularly advantageous in many-body systems, such as cold atoms, where the set of practical measurements is very limited and only variances of collective operators can typically be estimated. To show the practical relevance of our results, we derive simpler Schmidt-number criteria that require similar information as the fidelity-based witnesses, yet can detect a wider set of states. We also consider paradigmatic criteria based on spin covariances, which would be very helpful for experimental detection of high-dimensional entanglement in cold atom systems. We conclude by discussing the applicability of our results to a multiparticle ensemble and some open questions for future work.

arXiv:2208.04909•Aug 9, 2022•0 citations

Quantifying entanglement with covariance matrices

Oleg Gittsovich, Otfried Gühne

Covariance matrices are a useful tool for investigating correlations and entanglement in quantum systems. They are widely used in continuous-variable systems, but recently also for finite-dimensional systems powerful entanglement criteria in terms of covariance matrices have been derived. We show how these results can be used for the quantification of entanglement in bipartite systems. To that aim we introduce an entanglement parameter that quantifies the violation of the covariance matrix criterion and can be used to give lower bounds on the concurrence. These lower bounds are easily computable and give entanglement estimates for many weakly entangled states.

arXiv:0912.3018•Physical Review A•Mar 29, 2010•45 citations
We review an approach to the separability problem using covariance matrices of locally measurable observables. This approach leads to a strong entanglement criterion which we call covariance matrix criterion (CMC). We consider some corollaries of the CMC that in general allow to detect many bound entangled states. In particular we give a new example which sheds some light on the connections between different entanglement criteria derived from the general CMC.
We present a framework for deciding whether a quantum state is separable or entangled using covariance matrices of locally measurable observables. This leads to the covariance matrix criterion as a general separability criterion. We demonstrate that this criterion allows us to detect many states where the familiar criterion of the positivity of the partial transpose fails. It turns out that a large number of criteria that have been proposed to complement the positive partial transpose criterion---the computable cross norm or realignment criterion, the criterion based on local uncertainty relations, criteria derived from extensions of the realignment map, and others---are in fact a corollary of the covariance matrix criterion.
We propose a unifying approach to the separability problem using covariance matrices of locally measurable observables. From a practical point of view, our approach leads to strong entanglement criteria that allow us to detect the entanglement of many bound entangled states in higher dimensions and which are at the same time necessary and sufficient for two qubits. From a fundamental perspective, our approach leads to insights into the relations between several known entanglement criteria--such as the computable cross-norm and local uncertainty criteria--as well as their limitations.
We present a criterion for multiparticle entanglement based on covariance matrices. On the one hand, the criterion allows detection of bound entangled states which are not detected by other criteria; on the other hand, some strongly entangled pure states such as the Greenberger-Horne-Zeilinger (GHZ) states are not detected. We show, however, that this is a general phenomenon: No separability criterion based on two-particle correlations can recognize the entanglement in the family of graph states to which the GHZ states belong.
Entanglement plays a key role in quantum information. The separability problem arises: how to prove that a given quantum state is entangled? Covariance-matrix-based entanglement criteria are the most important and widely used approach to this problem in continuous variable systems. Here, the separability problem is systematically studied in the more experimentally realistic setting that the questioned state is only partly known. A simple general formalism is proposed for this variant of the separability problem. It builds relations between certain classes of criteria and bounds the power of these criteria. In particular, it shows the equivalence of many seemingly different covariance-matrix-based criteria; it offers guidance on the study of and reveals the inherent limitations of covariance-matrix-based criteria. Also, a Shannon entropic criterion is proposed for m ⊕ n-mode states. It unifies and improves many known covariance-matrix-based criteria and Shannon entropic criteria; it is necessary and sufficient for 1 ⊕ n-mode Gaussian states and m ⊕ n-mode bisymmetric Gaussian states.
High-dimensional entanglement has been identified as an important resource in quantum information processing, and also as a main obstacle for simulating quantum systems. Its certification is often difficult, and most widely used methods for experiments are based on fidelity measurements with respect to highly entangled states. Here, instead, we consider covariances of collective observables, as in the well-known Covariance Matrix Criterion (CMC)[1] and present a generalization of the CMC for determining the Schmidt number of a bipartite system. This is potentially particularly advantageous in many-body systems, such as cold atoms, where the set of practical measurements is very limited and only variances of collective operators can typically be estimated. To show the practical relevance of our results, we derive simpler Schmidt-number criteria that require similar information as the fidelity-based witnesses, yet can detect a wider set of states. We also consider paradigmatic criteria based on spin covariances, which would be very helpful for experimental detection of high-dimensional entanglement in cold atom systems. We conclude by discussing the applicability of our results to a multiparticle ensemble and some open questions for future work.
Covariance matrices are a useful tool for investigating correlations and entanglement in quantum systems. They are widely used in continuous-variable systems, but recently also for finite-dimensional systems powerful entanglement criteria in terms of covariance matrices have been derived. We show how these results can be used for the quantification of entanglement in bipartite systems. To that aim we introduce an entanglement parameter that quantifies the violation of the covariance matrix criterion and can be used to give lower bounds on the concurrence. These lower bounds are easily computable and give entanglement estimates for many weakly entangled states.