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Chinese Control Conference (CCC)

Bipartite entanglement of a bound entangled Gaussian state via a linear matrix inequality method*

Shan Ma, Lihan Zhou, Junfeng Ma, Shibei Xue

Published Jul 28, 2024• 0 citations
Publisher

Abstract

A four-mode Gaussian state has two types of bipartitions: 1-3 partitions and 2-2 partitions. A 1-3 partition means that a single mode forms one subsystem and the remaining three modes form the other subsystem; A 2-2 partition means that two modes form one subsystem and the remaining two modes form the other subsystem. In this paper, we investigate the entanglement property of a four-mode bound entangled Gaussian state for different bipartitions via a linear matrix inequality method. The Gaussian state example under consideration was proposed by R. F. Werner and M. M. Wolf in [1]. The results could be helpfu for deepening our understanding of Gaussian entanglement.

DOI
10.23919/ccc63176.2024.10661902

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A four-mode Gaussian state has two types of bipartitions: 1-3 partitions and 2-2 partitions. A 1-3 partition means that a single mode forms one subsystem and the remaining three modes form the other subsystem; A 2-2 partition means that two modes form one subsystem and the remaining two modes form the other subsystem. In this paper, we investigate the entanglement property of a four-mode bound entangled Gaussian state for different bipartitions via a linear matrix inequality method. The Gaussian state example under consideration was proposed by R. F. Werner and M. M. Wolf in [1]. The results could be helpfu for deepening our understanding of Gaussian entanglement.
We address the decomposition of a multimode pure Gaussian state with respect to a bipartite division of the modes. For any such division the state can always be expressed as a product state involving entangled two-mode squeezed states and single-mode local states at each side. The character of entanglement of the state can therefore be understood modewise; that is, a given mode on one side is entangled with only one corresponding mode of the other, and therefore the total bipartite entanglement is the sum of the modewise entanglement. This decomposition is generally not applicable to all mixed Gaussian states. However, the result can be extended to a special family of ``isotropic'' states, characterized by a phase space covariance matrix with a completely degenerate symplectic spectrum.
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.
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We introduce a Gaussian version of the entanglement of formation adapted to bipartite Gaussian states by considering decompositions into pure Gaussian states only. We show that this quantity is an entanglement monotone under Gaussian operations and provide a simplified computation for states of arbitrary many modes. For the case of one mode per site the remaining variational problem can be solved analytically. If the considered state is in addition symmetric with respect to interchanging the two modes, we prove additivity of the considered entanglement measure. Moreover, in this case and considering only a single copy, our entanglement measure coincides with the true entanglement of formation.
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We consider generic $(m\ifmmode\times\else\texttimes\fi{}n)$-mode bipartitions of continuous-variable systems, and study the associated bisymmetric multimode Gaussian states. They are defined as (m+n)(m+n)(m+n)-mode Gaussian states invariant under local mode permutations on the mmm-mode and nnn-mode subsystems. We prove that such states are equivalent, under local unitary transformations, to the tensor product of a two-mode state and of $m+n\ensuremath{-}2$ uncorrelated single-mode states. The entanglement between the mmm-mode and the nnn-mode blocks can then be completely concentrated on a single pair of modes by means of local unitary operations alone. This result allows us to prove that the PPT (positivity of the partial transpose) condition is necessary and sufficient for the separability of (m+n)(m+n)(m+n)-mode bisymmetric Gaussian states. We determine exactly their negativity and identify a subset of bisymmetric states whose multimode entanglement of formation can be computed analytically. We consider explicit examples of pure and mixed bisymmetric states and study their entanglement scaling with the number of modes.