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arXiv:1310.6470 · Physical Review A

Minkowski structure for purity and entanglement of Gaussian bipartite states

M. C. de Oliveira, F. Nicacio, Salomon Sylvain Mizrahi

Published Nov 20, 2013• 4 citations
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Abstract

The relation between the symplectic and Lorentz groups is explored to investigate entanglement features in a two-mode bipartite Gaussian state. We verify that the correlation matrix of arbitrary Gaussian states can be associated with a hyperbolic space with a Minkowski metric, which is divided in two regions: separabilitylike and entanglementlike, in equivalence to timelike and spacelike in special relativity. This correspondence naturally allows the definition of two insightful invariant squared distance measures: one related to the purity and another related to amount of entanglement. The second distance allows us to define a measure for entanglement in terms of the invariant interval between the given state and its closest separable state, given in a natural manner without the requirement of a minimization procedure.

DOI
10.1103/physreva.88.052324

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The relation between the symplectic and Lorentz groups is explored to investigate entanglement features in a two-mode bipartite Gaussian state. We verify that the correlation matrix of arbitrary Gaussian states can be associated with a hyperbolic space with a Minkowski metric, which is divided in two regions: separabilitylike and entanglementlike, in equivalence to timelike and spacelike in special relativity. This correspondence naturally allows the definition of two insightful invariant squared distance measures: one related to the purity and another related to amount of entanglement. The second distance allows us to define a measure for entanglement in terms of the invariant interval between the given state and its closest separable state, given in a natural manner without the requirement of a minimization procedure.
We present a geometric approach to the characterization of separability and entanglement in pure Gaussian states of an arbitrary number of modes. The analysis is performed adapting to continuous variables a formalism based on single subsystem unitary transformations that has been recently introduced to characterize separability and entanglement in pure states of qubits and qutrits [S. M. Giampaolo and F. Illuminati, Phys. Rev. A 76, 042301 (2007)]. In analogy with the finite-dimensional case, we demonstrate that the $1\ifmmode\times\else\texttimes\fi{}M$ bipartite entanglement of a multimode pure Gaussian state can be quantified by the minimum squared Euclidean distance between the state itself and the set of states obtained by transforming it via suitable local symplectic (unitary) operations. This minimum distance, corresponding to aaa, uniquely determined, extremal local operation, defines an entanglement monotone equivalent to the entropy of entanglement, and amenable to direct experimental measurement with linear optical schemes.
We introduce a locally symplectic-invariant quantifier of correlations between arbitrary bosonic Gaussian modes, with particular emphasis on quantum field theory. The quantity, denoted by~Dsym\mathcal{D}^{\mathrm{sym}}Dsym, admits a simple geometric interpretation as the symmetric overlap between each mode and the purification partner of the other, providing a direct geometric characterization of how correlations are distributed between modes. We derive a necessary and sufficient criterion for two-mode Gaussian entanglement in terms of Dsym\mathcal{D}^{\mathrm{sym}}Dsym, placing on firm quantitative footing the intuition that the entanglement of a localized mode is encoded in the spatial support of its purification partner. We demonstrate the framework for wavepacket modes of a scalar quantum field in Minkowski spacetime (illustrating how the geometry of partner modes reveals the spatial structure of quantum correlations) and discuss extensions to multimode systems and mixed Gaussian states.
The entanglement properties of a two-qubit system are analyzed from the point of view of Lie algebras and geometry. We deal with $4\ifmmode\times\else\texttimes\fi{}4$ matrices belonging to a particular class having seven free parameters, constructed from the imposed symmetries on two-qubit states expanded in terms of a basis whose elements belong to a subalgebra of su(4). The entanglement or separability character of the state is determined and measured by analyzing the eigenvalues of an auxiliary matrix obtained from the original state after performing on it a nonunitary operation. In terms of its eigenvalues, we define two squared distances having a Minkowski metric, ${s}^{2}={t}^{2}\ensuremath{-}{\stackrel{P\vec}{V}}^{2}$, which become a measure of the entanglement. The Minkowski metric is a signature of that class of two-qubit states (matrices) and also of the auxiliary matrices to be constructed. The squared distance is invariant by any unitary transformation on the two-qubit state. We illustrate the theory with examples.
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.
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.
Abstract We investigate different geometries and invariant measures on the space of mixed Gaussian quantum states. We show that when the global purity of the state is held fixed, these measures coincide and it is possible, within this constraint, to define a unique notion of volume on the space of mixed Gaussian states. We then use the so defined measure to study typical non-classical correlations of two mode mixed Gaussian quantum states, in particular entanglement and steerability. We show that under the purity constraint alone, typical values for symplectic invariants can be computed very elegantly, irrespectively of the non-compactness of the underlying state space. Then we consider finite volumes by constraining the purity and energy of the Gaussian state and compute typical values of quantum correlations numerically.