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

Journal of Physics A Mathematical and Theoretical

Multipartite Gaussian entanglement measure with applications to graph states and bosonic field theory

Matteo Gori, Matthieu Sarkis, Alexandre Tkatchenko

Published Nov 19, 2025• 0 citations
Publisher

Abstract

Abstract Computationally feasible multipartite entanglement measures are essential for advancing our understanding of complex quantum systems. Entanglement distance (ED), introduced by Cocchiarella et al (2020 Phys. Rev. A 101 042129), based on the Fubini–Study metric, offers several advantages over existing methods, including ease of computation, a profound geometrical interpretation, and applicability to multipartite entanglement. Although ED has been successfully applied to systems of qudits, an explicit formulation for continuous quantum variables, particularly for pure Gaussian states, remains unexplored. In this work, we address this limitation by deriving the analytical expression for the Gaussian entanglement measure (GEM), a multipartite entanglement monotone for multimode pure Gaussian states based on the purity of fragments of the whole system, through a generalization of ED to group-theoretic coherent states. We show the efficacy of GEM across various scenarios, including the analysis of a two-mode Gaussian state under beam-splitter and squeezing transformations, and exploring graph states involving three and four modes. Notably, comparing GEM values for states with different graph topologies reveals insights into the connectivity of the underlying graphs. Additionally, we illustrate how GEM provides insights into free bosonic field theory on R t × S 1 beyond standard bipartite entanglement entropy, paving the way for quantum information-theoretic tools to probe the topological properties of quantum field theories.

DOI
10.1088/1751-8121/ae21a9

Related papers

Gaussian Entanglement Measure: Applications to Multipartite Entanglement of Graph States and Bosonic Field Theory

Matteo Gori, Matthieu Sarkis, Alexandre Tkatchenko

Computationally feasible multipartite entanglement measures are needed to advance our understanding of complex quantum systems. An entanglement measure based on the Fubini-Study metric has been recently introduced by Cocchiarella and co-workers, showing several advantages over existing methods, including ease of computation, a deep geometrical interpretation, and applicability to multipartite entanglement. Here, we present the Gaussian Entanglement Measure (GEM), a generalization of geometric entanglement measure for multimode Gaussian states, based on the purity of fragments of the whole systems. Our analysis includes the application of GEM to a two-mode Gaussian state coupled through a combined beamsplitter and a squeezing transformation. Additionally, we explore 3-mode and 4-mode graph states, where each vertex represents a bosonic mode, and each edge represents a quadratic transformation for various graph topologies. Interestingly, the ratio of the geometric entanglement measures for graph states with different topologies naturally captures properties related to the connectivity of the underlying graphs. Finally, by providing a computable multipartite entanglement measure for systems with a large number of degrees of freedom, we show that our definition can be used to obtain insights into a free bosonic field theory on $\mathbb R_t\times S^1$, going beyond the standard bipartite entanglement entropy approach between different regions of spacetime. The results presented herein suggest how the GEM paves the way for using quantum information-theoretical tools to study the topological properties of the space on which a quantum field theory is defined.

arXiv:2401.17938•Jan 31, 2024•1 citation

Geometric Aspects of Entanglement

Lucio De Simone, Lorenzo Capra, Arthur Vesperini +3 more

Quantum entanglement is a fundamental resource in quantum information theory, yet its general characterization and quantification remain challenging, especially in multipartite systems. In this work we investigate entanglement from a geometric perspective, focusing on the Riemannian structure induced by the Fubini--Study metric on the projective Hilbert space of multi-qubit quantum states. By exploiting the local-unitary invariance of this metric, we derive the entanglement distance (ED), a geometric measure that quantifies entanglement as an obstruction to locally minimizing the sum of squared Fubini--Study distances generated by local operations. We analyze the properties of ED for pure multi-qubit states and discuss its behavior under local operations and classical communication. In particular, we show that ED reproduces established entanglement measures in well-defined and restricted settings. For pure states of two qubits, ED reduces to an exact monotone function of the concurrence and, independently, to an explicit monotone function of the entropy of entanglement. These results provide a clear geometric interpretation of standard bipartite entanglement measures within the present framework, while highlighting the limitations of such correspondences beyond the two-qubit case.

quant-pharXiv:2610.03446•Entropy•Oct 5, 2026•1 citation

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

A Geometric Theory of Quantum Entanglement

Lorenzo Capra, Lucio de Simone, Roberto Franzosi

Entanglement Distance (ED) was originally proposed as a geometric measure of entanglement derived from the Fubini-Study metric on the projective Hilbert space. Independently, the Meyer-Wallach and Scott measures quantify multipartite entanglement via linear entropy. In this work, we demonstrate that these two seemingly distinct frameworks are mathematically identical for pure states of arbitrary finite dimensions. We prove that ED arises naturally as the trace of the Fubini-Study metric tensor over the local subalgebra of observables. Crucially, this geometric unification yields a direct operational interpretation: the global entanglement of a pure state is exactly proportional to the total Quantum Fisher Information (QFI) available for local unitary estimation. This bridges abstract information geometry with quantum metrology, demonstrating that ED dynamically quantifies resourcefulness for distributed quantum sensing, identifying Heisenberg-limited sensitivity in regimes where standard variance-based witnesses fail.

quant-pharXiv:2609.11720•Sep 11, 2026•0 citations

Bipartite and Multipartite Entanglement of Gaussian States

Gerardo Adesso, Fabrizio Illuminati

In this chapter we review the characterization of entanglement in Gaussian states of continuous variable systems. For two-mode Gaussian states, we discuss how their bipartite entanglement can be accurately quantified in terms of the global and local amounts of mixedness, and efficiently estimated by direct measurements of the associated purities. For multimode Gaussian states endowed with local symmetry with respect to a given bipartition, we show how the multimode block entanglement can be completely and reversibly localized onto a single pair of modes by local, unitary operations. We then analyze the distribution of entanglement among multiple parties in multimode Gaussian states. We introduce the continuous-variable tangle to quantify entanglement sharing in Gaussian states and we prove that it satisfies the Coffman-Kundu-Wootters monogamy inequality. Nevertheless, we show that pure, symmetric three-mode Gaussian states, at variance with their discrete-variable counterparts, allow a promiscuous sharing of quantum correlations, exhibiting both maximum tripartite residual entanglement and maximum couplewise entanglement between any pair of modes. Finally, we investigate the connection between multipartite entanglement and the optimal fidelity in a continuous-variable quantum teleportation network. We show how the fidelity can be maximized in terms of the best preparation of the shared entangled resources and, viceversa, that this optimal fidelity provides a clearcut operational interpretation of several measures of bipartite and multipartite entanglement, including the entanglement of formation, the localizable entanglement, and the continuous-variable tangle.

arXiv:quant-ph/0510052•IMPERIAL COLLEGE PRESS eBooks•Feb 1, 2007•10 citations

Multipartite entanglement in three-mode Gaussian states of continuous-variable systems: Quantification, sharing structure, and decoherence

Gerardo Adesso, Alessio Serafini, Fabrizio Illuminati

We present a complete analysis of the multipartite entanglement of three-mode Gaussian states of continuous-variable systems. We derive standard forms which characterize the covariance matrix of pure and mixed three-mode Gaussian states up to local unitary operations, showing that the local entropies of pure Gaussian states are bound to fulfill a relationship which is stricter than the general Araki-Lieb inequality. Quantum correlations can be quantified by a proper convex roof extension of the squared logarithmic negativity, the continuous-variable tangle, or contangle. We review and elucidate in detail the proof that in multimode Gaussian states the contangle satisfies a monogamy inequality constraint [G. Adesso and F. Illuminati, New J. Phys8, 15 (2006)]. The residual contangle, emerging from the monogamy inequality, is an entanglement monotone under Gaussian local operations and classical communications and defines a measure of genuine tripartite entanglements. We determine the analytical expression of the residual contangle for arbitrary pure three-mode Gaussian states and study in detail the distribution of quantum correlations in such states. This analysis yields that pure, symmetric states allow for a promiscuous entanglement sharing, having both maximum tripartite entanglement and maximum couplewise entanglement between any pair of modes. We thus name these states $\mathrm{GHZ}∕W$ states of continuous-variable systems because they are simultaneous continuous-variable counterparts of both the GHZ and the $W$ states of three qubits. We finally consider the effect of decoherence on three-mode Gaussian states, studying the decay of the residual contangle. The $\mathrm{GHZ}∕W$ states are shown to be maximally robust against losses and thermal noise.

arXiv:quant-ph/0512124•Physical Review A•Mar 30, 2006•220 citations
Abstract Computationally feasible multipartite entanglement measures are essential for advancing our understanding of complex quantum systems. Entanglement distance (ED), introduced by Cocchiarella et al (2020 Phys. Rev. A 101 042129), based on the Fubini–Study metric, offers several advantages over existing methods, including ease of computation, a profound geometrical interpretation, and applicability to multipartite entanglement. Although ED has been successfully applied to systems of qudits, an explicit formulation for continuous quantum variables, particularly for pure Gaussian states, remains unexplored. In this work, we address this limitation by deriving the analytical expression for the Gaussian entanglement measure (GEM), a multipartite entanglement monotone for multimode pure Gaussian states based on the purity of fragments of the whole system, through a generalization of ED to group-theoretic coherent states. We show the efficacy of GEM across various scenarios, including the analysis of a two-mode Gaussian state under beam-splitter and squeezing transformations, and exploring graph states involving three and four modes. Notably, comparing GEM values for states with different graph topologies reveals insights into the connectivity of the underlying graphs. Additionally, we illustrate how GEM provides insights into free bosonic field theory on R t × S 1 beyond standard bipartite entanglement entropy, paving the way for quantum information-theoretic tools to probe the topological properties of quantum field theories.
Computationally feasible multipartite entanglement measures are needed to advance our understanding of complex quantum systems. An entanglement measure based on the Fubini-Study metric has been recently introduced by Cocchiarella and co-workers, showing several advantages over existing methods, including ease of computation, a deep geometrical interpretation, and applicability to multipartite entanglement. Here, we present the Gaussian Entanglement Measure (GEM), a generalization of geometric entanglement measure for multimode Gaussian states, based on the purity of fragments of the whole systems. Our analysis includes the application of GEM to a two-mode Gaussian state coupled through a combined beamsplitter and a squeezing transformation. Additionally, we explore 3-mode and 4-mode graph states, where each vertex represents a bosonic mode, and each edge represents a quadratic transformation for various graph topologies. Interestingly, the ratio of the geometric entanglement measures for graph states with different topologies naturally captures properties related to the connectivity of the underlying graphs. Finally, by providing a computable multipartite entanglement measure for systems with a large number of degrees of freedom, we show that our definition can be used to obtain insights into a free bosonic field theory on Rt×S1\mathbb R_t\times S^1Rt​×S1, going beyond the standard bipartite entanglement entropy approach between different regions of spacetime. The results presented herein suggest how the GEM paves the way for using quantum information-theoretical tools to study the topological properties of the space on which a quantum field theory is defined.
Quantum entanglement is a fundamental resource in quantum information theory, yet its general characterization and quantification remain challenging, especially in multipartite systems. In this work we investigate entanglement from a geometric perspective, focusing on the Riemannian structure induced by the Fubini--Study metric on the projective Hilbert space of multi-qubit quantum states. By exploiting the local-unitary invariance of this metric, we derive the entanglement distance (ED), a geometric measure that quantifies entanglement as an obstruction to locally minimizing the sum of squared Fubini--Study distances generated by local operations. We analyze the properties of ED for pure multi-qubit states and discuss its behavior under local operations and classical communication. In particular, we show that ED reproduces established entanglement measures in well-defined and restricted settings. For pure states of two qubits, ED reduces to an exact monotone function of the concurrence and, independently, to an explicit monotone function of the entropy of entanglement. These results provide a clear geometric interpretation of standard bipartite entanglement measures within the present framework, while highlighting the limitations of such correspondences beyond the two-qubit case.
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.
Entanglement Distance (ED) was originally proposed as a geometric measure of entanglement derived from the Fubini-Study metric on the projective Hilbert space. Independently, the Meyer-Wallach and Scott measures quantify multipartite entanglement via linear entropy. In this work, we demonstrate that these two seemingly distinct frameworks are mathematically identical for pure states of arbitrary finite dimensions. We prove that ED arises naturally as the trace of the Fubini-Study metric tensor over the local subalgebra of observables. Crucially, this geometric unification yields a direct operational interpretation: the global entanglement of a pure state is exactly proportional to the total Quantum Fisher Information (QFI) available for local unitary estimation. This bridges abstract information geometry with quantum metrology, demonstrating that ED dynamically quantifies resourcefulness for distributed quantum sensing, identifying Heisenberg-limited sensitivity in regimes where standard variance-based witnesses fail.
In this chapter we review the characterization of entanglement in Gaussian states of continuous variable systems. For two-mode Gaussian states, we discuss how their bipartite entanglement can be accurately quantified in terms of the global and local amounts of mixedness, and efficiently estimated by direct measurements of the associated purities. For multimode Gaussian states endowed with local symmetry with respect to a given bipartition, we show how the multimode block entanglement can be completely and reversibly localized onto a single pair of modes by local, unitary operations. We then analyze the distribution of entanglement among multiple parties in multimode Gaussian states. We introduce the continuous-variable tangle to quantify entanglement sharing in Gaussian states and we prove that it satisfies the Coffman-Kundu-Wootters monogamy inequality. Nevertheless, we show that pure, symmetric three-mode Gaussian states, at variance with their discrete-variable counterparts, allow a promiscuous sharing of quantum correlations, exhibiting both maximum tripartite residual entanglement and maximum couplewise entanglement between any pair of modes. Finally, we investigate the connection between multipartite entanglement and the optimal fidelity in a continuous-variable quantum teleportation network. We show how the fidelity can be maximized in terms of the best preparation of the shared entangled resources and, viceversa, that this optimal fidelity provides a clearcut operational interpretation of several measures of bipartite and multipartite entanglement, including the entanglement of formation, the localizable entanglement, and the continuous-variable tangle.
We present a complete analysis of the multipartite entanglement of three-mode Gaussian states of continuous-variable systems. We derive standard forms which characterize the covariance matrix of pure and mixed three-mode Gaussian states up to local unitary operations, showing that the local entropies of pure Gaussian states are bound to fulfill a relationship which is stricter than the general Araki-Lieb inequality. Quantum correlations can be quantified by a proper convex roof extension of the squared logarithmic negativity, the continuous-variable tangle, or contangle. We review and elucidate in detail the proof that in multimode Gaussian states the contangle satisfies a monogamy inequality constraint [G. Adesso and F. Illuminati, New J. Phys8, 15 (2006)]. The residual contangle, emerging from the monogamy inequality, is an entanglement monotone under Gaussian local operations and classical communications and defines a measure of genuine tripartite entanglements. We determine the analytical expression of the residual contangle for arbitrary pure three-mode Gaussian states and study in detail the distribution of quantum correlations in such states. This analysis yields that pure, symmetric states allow for a promiscuous entanglement sharing, having both maximum tripartite entanglement and maximum couplewise entanglement between any pair of modes. We thus name these states GHZ∕W\mathrm{GHZ}∕WGHZ∕W states of continuous-variable systems because they are simultaneous continuous-variable counterparts of both the GHZ and the WWW states of three qubits. We finally consider the effect of decoherence on three-mode Gaussian states, studying the decay of the residual contangle. The GHZ∕W\mathrm{GHZ}∕WGHZ∕W states are shown to be maximally robust against losses and thermal noise.