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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Spatio-temporal properties of multipartite entanglement

Giuseppe Patera, Mikhail I. Kolobov

Published Apr 30, 2010• 0 citations
Publisher

Abstract

Quantum multipartite entanglement is a striking phenomenon predicted by quantum mechanics when several parts of a physical system share the same quantum state that cannot be factorized into the states of individual subsystems. The Gaussian quantum states are usually characterized by the covariance matrix of the quadrature components. A powerful formalism for treating the Gaussian states is that of the symplectic eigenvalues. In particular, a quantitative measure of multipartite entanglement is the so-called logarithmic negativity, related to the symplectic eigenvalues of the partially transposed covariance matrix. Considering only global variances of the field quadratures one completely neglects the spatiotemporal properties of the electromagnetic field. We propose, following the spirit of quantum imaging, to generalize the theory of multipartite entanglement for the continuous variable Gaussian states by considering the local correlation matrix at two different spatiotemporal points [see manuscript for characters] and [see manuscript for characters] with [see manuscript for characters] being the transverse coordinate. For stationary and homogeneous systems one can also introduce the spatiotemporal Fourier components of the correlation matrix. The formalism of the global symplectic eigenvalues can be straightforwardly generalized to the frequency-dependent symplectic eigenvalues. This generalized theory allows, in particular, to introduce the characteristic spatial area and time of the multipartite entanglement, which in general depend on the number of "parties" in the system. As an example we consider multipartite entanglement in spontaneous parametric down-conversion with spatially-structured pump. We investigate spatial properties of such entanglement and calculate its characteristic spatial length.

DOI
10.1117/12.853777

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Quantum multipartite entanglement is a striking phenomenon predicted by quantum mechanics when several parts of a physical system share the same quantum state that cannot be factorized into the states of individual subsystems. The Gaussian quantum states are usually characterized by the covariance matrix of the quadrature components. A powerful formalism for treating the Gaussian states is that of the symplectic eigenvalues. In particular, a quantitative measure of multipartite entanglement is the so-called logarithmic negativity, related to the symplectic eigenvalues of the partially transposed covariance matrix. Considering only global variances of the field quadratures one completely neglects the spatiotemporal properties of the electromagnetic field. We propose, following the spirit of quantum imaging, to generalize the theory of multipartite entanglement for the continuous variable Gaussian states by considering the local correlation matrix at two different spatiotemporal points [see manuscript for characters] and [see manuscript for characters] with [see manuscript for characters] being the transverse coordinate. For stationary and homogeneous systems one can also introduce the spatiotemporal Fourier components of the correlation matrix. The formalism of the global symplectic eigenvalues can be straightforwardly generalized to the frequency-dependent symplectic eigenvalues. This generalized theory allows, in particular, to introduce the characteristic spatial area and time of the multipartite entanglement, which in general depend on the number of "parties" in the system. As an example we consider multipartite entanglement in spontaneous parametric down-conversion with spatially-structured pump. We investigate spatial properties of such entanglement and calculate its characteristic spatial length.
In this Rapid Communication, we propose, following the spirit of quantum imaging, to generalize the theory of multipartite entanglement for the continuous-variable Gaussian states by considering, instead of the global covariance matrix, the local correlation matrix at two different spatiotemporal points $(\mathrm{\ensuremath{\rho}P\vec},t)$ and $(\mathrm{\ensuremath{\rho}P\vec} {}^{'},{t}^{'})$, with $\mathrm{\ensuremath{\rho}P\vec}$ being the transverse coordinate. Our approach makes it possible to introduce the characteristic spatial length and the characteristic time of the multipartite entanglement, which in general depend on the number of ``parties'' in the system. As an example, we consider tripartite entanglement in spontaneous parametric down-conversion with a spatially structured pump. We investigate spatiotemporal properties of such entanglement and calculate its characteristic spatial length and time.
Is entanglement an exclusive feature of quantum systems, or is it common to all non-classical theories? And if this is the case, how strong is quantum mechanical entanglement as compared to that exhibited by other theories? The first part of this thesis deals with these questions by considering quantum theory as part of a wider landscape of physical theories, collectively called general probabilistic theories (GPTs). Among the other things, this manuscript contains a detailed introduction to the abstract state space formalism for GPTs. We start with a comprehensive review of the proof of a famous theorem by Ludwig that constitutes one of its cornerstones (Ch. 1). After explaining the basic rules of the game, we translate our questions into precise conjectures and present our progress toward a full solution (Ch. 2). In Ch. 3 we consider entanglement at the level of measurements instead of states, focusing on one of its main implications, i.e. data hiding. We determine the maximal data hiding strength that a quantum mechanical system can exhibit, and also the maximum value among all GPTs, finding that the former scales as the square root of the latter. In the second part of this manuscript we look into quantum entanglement. In Ch. 4 we discuss the entanglement transformation properties of a class of maps that model white noise acting either locally or globally on a bipartite system. In Ch. 5 we employ matrix analysis tools to develop a unified approach to Gaussian entanglement. The third part of this thesis concerns more general forms of non-classical correlations in bipartite continuous variable systems. In Ch. 6 we devise a general scheme that allows to consistently classify correlations of bipartite Gaussian states into classical and quantum ones. Finally, Ch. 7 explores some problems connected with a certain strong subadditivity matrix inequality.
Abstract In quantum information theory, the entropy of entanglement is a standard measure of bipartite entanglement between two partitions of a composite system. For a particular class of continuous variable quantum states, the Gaussian states, the entropy of entanglement can be expressed elegantly in terms of symplectic eigenvalues , elements that characterise a Gaussian state and depend on the correlations of the canonical variables. We give a rigorous step-by-step derivation of this result and provide physical insights, together with an example that can be useful in practice for calculations.
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.
We characterize the exact trade-off between internal and external entanglement in asymmetric two-mode Gaussian states at fixed local symplectic eigenvalues. Internal entanglement is quantified by the negativity NA:B\mathcal{N}_{A:B}NA:B​ between the two modes, while external entanglement is quantified by the negativity NAB:CD\mathcal{N}_{AB:CD}NAB:CD​ across a Gaussian purification. Numerical sampling suggests the boundary structure of the attainable region in the (NAB:CD,NA:B)(\mathcal{N}_{AB:CD},\mathcal{N}_{A:B})(NAB:CD​,NA:B​) plane, which we then determine analytically. Parametrizing the global symplectic spectrum yields a closed admissible interval for the remaining spectral parameter at fixed NAB:CD\mathcal{N}_{AB:CD}NAB:CD​. We prove constructively that every point in this interval corresponds to a physical covariance matrix, implying that all intermediate values of NA:B\mathcal{N}_{A:B}NA:B​ between the resulting sharp bounds are achievable. The upper bound is attained for opposite-sign correlations of equal magnitude, while the positive branch of the lower bound occurs when one global symplectic eigenvalue equals unity. We also identify two external-entanglement thresholds, derive a saturated monogamy-type inequality, and recover the symmetric-state limit. Because the external negativity depends only on the spectrum of the reduced two-mode state, the bounds hold for any purification, including non-Gaussian purifications.