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:2307.13480 · Entropy

Covariance-Matrix-Based Criteria for Network Entanglement

Kiara Hansenne, Otfried Gühne

Published Aug 24, 2023• 0 citations
arXivHTMLPDF

Abstract

Quantum networks offer a realistic and practical scheme for generating multiparticle entanglement and implementing multiparticle quantum communication protocols. However, the correlations that can be generated in networks with quantum sources and local operations are not yet well understood. Covariance matrices, which are powerful tools in entanglement theory, have been also applied to the network scenario. We present simple proofs for the decomposition of such matrices into the sum of positive semi-definite block matrices and, based on that, develop analytical and computable necessary criteria for preparing states in quantum networks. These criteria can be applied to networks where nodes share at most one source, such as all bipartite networks.

DOI
10.3390/e25091260

Related papers

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

Covariance Decomposition as a Universal Limit on Correlations in Networks

Salman Beigi, Marc-Olivier Renou

Parties connected to independent sources through a network can generate correlations among themselves. Notably, the space of feasible correlations for a given network, depends on the physical nature of the sources and the measurements performed by the parties. In particular, quantum sources give access to nonlocal correlations that cannot be generated classically. In this paper, we derive a universal limit on correlations in networks in terms of their covariance matrix. We show that in a network satisfying a certain condition, the covariance matrix of any feasible correlation can be decomposed as a summation of positive semidefinite matrices each of whose terms corresponds to a source in the network. Our result is universal in the sense that it holds inanyphysical theory of correlation in networks, including the classical, quantum and all generalized probabilistic theories.

arXiv:2103.14840•IEEE Transactions on Information Theory•Oct 14, 2021•14 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

Characterizing quantum networks: Insights from coherence theory

Tristan Kraft, Cornelia Spee, Xiao‐Dong Yu +1 more

Networks based on entangled quantum systems enable interesting applications in quantum information processing and the understanding of the resulting quantum correlations is essential for advancing the technology. We show that the theory of quantum coherence provides powerful tools for analyzing this problem. For that, we demonstrate that a recently proposed approach to network correlations based on covariance matrices can be improved and analytically evaluated for the most important cases.

arXiv:2006.06693•Physical Review A•May 5, 2021•30 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
Quantum networks offer a realistic and practical scheme for generating multiparticle entanglement and implementing multiparticle quantum communication protocols. However, the correlations that can be generated in networks with quantum sources and local operations are not yet well understood. Covariance matrices, which are powerful tools in entanglement theory, have been also applied to the network scenario. We present simple proofs for the decomposition of such matrices into the sum of positive semi-definite block matrices and, based on that, develop analytical and computable necessary criteria for preparing states in quantum networks. These criteria can be applied to networks where nodes share at most one source, such as all bipartite networks.
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.
Parties connected to independent sources through a network can generate correlations among themselves. Notably, the space of feasible correlations for a given network, depends on the physical nature of the sources and the measurements performed by the parties. In particular, quantum sources give access to nonlocal correlations that cannot be generated classically. In this paper, we derive a universal limit on correlations in networks in terms of their covariance matrix. We show that in a network satisfying a certain condition, the covariance matrix of any feasible correlation can be decomposed as a summation of positive semidefinite matrices each of whose terms corresponds to a source in the network. Our result is universal in the sense that it holds inanyphysical theory of correlation in networks, including the classical, quantum and all generalized probabilistic theories.
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.