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arXiv:quant-ph/0312071 · International Journal of Quantum Information

INTRODUCTION TO THE BASICS OF ENTANGLEMENT THEORY IN CONTINUOUS-VARIABLE SYSTEMS

Jens Eisert, Martin Bodo Plenio

Published Dec 1, 2003• 360 citations
arXivHTMLPDF

Abstract

We outline the basic questions that are being studied in any theory of entanglement. Following a brief review of some of the main achievements of entanglement theory for finite-dimensional systems such as qubits, we will then consider entanglement in infinite-dimensional systems. Asking for a theory of entanglement in such systems under experimentally feasible operations leads to the development of the theory of entanglement of Gaussian states. Results of this theory are presented and the tools that have been developed for this theory are then applied to a number of problems.

DOI
10.1142/s0219749903000371

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We outline the basic questions that are being studied in any theory of entanglement. Following a brief review of some of the main achievements of entanglement theory for finite-dimensional systems such as qubits, we will then consider entanglement in infinite-dimensional systems. Asking for a theory of entanglement in such systems under experimentally feasible operations leads to the development of the theory of entanglement of Gaussian states. Results of this theory are presented and the tools that have been developed for this theory are then applied to a number of problems.
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.
The study of Gaussian states has arisen to a privileged position in continuous variable quantum information in recent years. This is due to vehemently pursued experimental realisations and a magnificently elegant mathematical framework. In this paper, we provide a brief, and hopefully didactic, exposition of Gaussian state quantum information and its contemporary uses, including sometimes omitted crucial details. After introducing the subject material and outlining the essential toolbox of continuous variable systems, we define the basic notions needed to understand Gaussian states and Gaussian operations. In particular, emphasis is placed on the mathematical structure combining notions of algebra and symplectic geometry fundamental to a complete understanding of Gaussian informatics. Furthermore, we discuss the quantification of different forms of correlations (including entanglement and quantum discord) for Gaussian states, paying special attention to recently developed measures. The paper is concluded by succinctly expressing the main Gaussian state limitations and outlining a selection of possible future lines for quantum information processing with continuous variable systems.
G.A. RAGGIO Abstract. We give a brief and incomplete survey of the problem of entan- glement of of composite quantum systems. 1960's - 1980's there were serious attempts (all of them anticipated by J. von Neu- mann) to do away with the underlying Hilbert space and work directly with the abstract algebraic structure modeled by C � -algebras or W � -algebras (abstract von Neumann algebras)(1, 2, 3). This was particularly fruitful when dealing with sys- tems of infinitely many degrees of freedom (quantum fields, thermodynamic limits, etc.). The problem of entanglement which we want to address here can be formu- lated quite straightforwardly in this algebraic framework of quantum theory. But we will stick to the quantum mechanics of the 1930's and keep the Hilbert space. Mainly because the results which are available concern the finite dimensional case. The states of the quantum system specified by H are associated with the lin- ear functionals f : B(H) → C which are positive (i.e., f(a) ≥ 0 for all positive a ∈ B(H)), normalized (i.e., f(1) = 1) and normal (i.e., f(supn{an}) = supnf(an) for any increasing family of selfadjoint operators {an : n = 1,2, � � � } which is uni- formly bounded). It is this normality condition which guarantees that the are in one-to-one correspondence with positive trace class operators d of unit trace (density operators) via the formula B(H) ∋ a 7→ tr(da). This gives an extremely convenient representation of and one often confuses the state f as a linear functional with the associated density operator d for which f(a) = tr(da). States are automatically continuous, and satisfy f(a � ) = f(a). The complex number f(a) is interpreted probabilistically as the expected value of the observable associated with the operator a when the system is in the state f. The normalization condi- tion is thus necessary for the consistency of this interpretation and the normality condition is seen as a non-commutative version of the �-additivity of probability measures. Clearly states, which we will denote by �(H), form a convex set which is closed
This paper reviews recent advances in quantum learning theory for continuous-variable (CV) systems. Quantum learning theory investigates how to extract classical information from quantum systems as efficiently as possible. CV systems are ubiquitous in nature and in quantum technologies, as they describe bosonic and quantum-optical systems. While quantum learning theory for finite-dimensional systems has been extensively studied, the corresponding theory for CV systems has only recently begun to develop; here we provide a concise review. We focus on the following questions: what is the minimum number of copies (the sample complexity) required to learn a non-Gaussian state, possibly under energy constraints? What is the sample complexity for learning Gaussian states? How does the performance of CV state learning depend on non-Gaussianity? How can one test whether a state is Gaussian or far from the set of Gaussian states? And how can Gaussian processes be learned efficiently? Central to these topics, we also review several bounds on the trace distance between CV states in terms of their covariance matrices, which may be of independent interest. Overall, this work summarises selected developments in tomography of CV systems and highlights a selection of open problems.
We consider the case of a pair of particles initially in a superposition state corresponding to a separated pair of wave packets. In contrast to a previous related work, we avoid a master equation approach and we calculate exactly the time development of this non-Gaussian state due to interaction with an arbitrary heat bath. We find that coherence decays continuously, as expected. We then investigate entanglement and find that at a finite time the system becomes separable (not entangled). Thus, we see that entanglement sudden death is also prevalent in continuous variable systems which should raise concern for the designers of entangled systems.