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arXiv:quant-ph/0504163 · Quantum Information and Computation

An introduction to entanglement measures

Martin Bodo Plenio, S. Virmani

Published Jan 1, 2007• 754 citations
arXivHTMLPDF

Abstract

We review the theory of entanglement measures, concentrating mostly on the finite dimensional two-party case. Topics covered include: single-copy and asymptotic entanglement manipulation; the entanglement of formation; the entanglement cost; the distillable entanglement; the relative entropic measures; the squashed entanglement; log-negativity; the robustness monotones; the greatest cross-norm; uniqueness and extremality theorems. Infinite dimensional systems and multi-party settings will be discussed briefly.

DOI
10.26421/qic7.1-2-1

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Abstract In this chapter the quantification of entanglement is discussed. Beginning with the pure-state entanglement measure called the entropy of entanglement, we discuss the paradigm of local operations and classical communication (LOCC) and its relation to majorization via Nielsen’s majorization theorem. We then turn to the asymptotic setting and discuss the formation and distillation of entanglement and the related entanglement measures: entanglement cost and distillable entanglement, and we examine the notions of distillability and bound entanglement. This brings us to a more general discussion of entanglement measures and monotones, and their desired properties, during which we present the entanglement of formation and concurrence, squashed entanglement, as well as the tangle and its relation to monogamy of entanglement, but also quantities like the relative entropy of entanglement, the Hilbert-Schmidt measure, and the (logarithmic) negativity. Finally we turn to the construction of entanglement witnesses and their geometric interpretation via the Bertlmann-Narnhofer-Thirring theorem
We explore and develop the mathematics of the theory of entanglement measures. After a careful review and analysis of definitions, of preliminary results, and of connections between conditions on entanglement measures, we prove a sharpened version of a uniqueness theorem which gives necessary and sufficient conditions for an entanglement measure to coincide with the reduced von Neumann entropy on pure states. We also prove several versions of a theorem on extreme entanglement measures in the case of mixed states. We analyze properties of the asymptotic regularization of entanglement measures proving, for example, convexity for the entanglement cost and for the regularized relative entropy of entangle ment.
We study the robustness of asymptotic entanglement manipulation for Mazzola--Sutter--Renner (MSR) almost-iid sources, which allow a sublinear number of deviations from tensor-power structure. For pure sources along ∣φ⟩AB|φ\rangle_{AB}∣φ⟩AB​, every concentration rate below S(φA)S(φ_A)S(φA​) is achievable by a single Schur--Weyl protocol depending only on the rate and local Hilbert spaces. For general sources along ρABρ_{AB}ρAB​, every distillation rate below ED(ρAB)E_D(ρ_{AB})ED​(ρAB​) is achievable by a single LOCC protocol sequence independent of the particular source and defect size. Its error vanishes exponentially and uniformly for each prescribed defect bound rn=o(n)r_n=o(n)rn​=o(n), yielding an optimal universal distillation rate equal to ED(ρAB)E_D(ρ_{AB})ED​(ρAB​). For dilution, the entanglement cost of every MSR almost-iid target sequence along ρABρ_{AB}ρAB​ is at most EF∞(ρAB)E_F^\infty(ρ_{AB})EF∞​(ρAB​), the regularized entanglement of formation of the reference state. Consequently, every MSR almost-iid sequence along a pure reference state ∣φ⟩AB|φ\rangle_{AB}∣φ⟩AB​ is asymptotically reversible, with distillable entanglement and entanglement cost equal to S(φA)S(φ_A)S(φA​), even when its individual states are mixed. The proofs establish structural and entropic properties of MSR almost-iid sources that may be useful in other information-theoretic settings.
Measures of entanglement can be employed for the analysis of numerous quantum information protocols. Due to computational convenience, logarithmic negativity is often the choice in the case of continuous-variable systems. In this work, we analyze a continuous-variable measurement-based entanglement distillation experiment using a collection of entanglement measures. This includes logarithmic negativity, entanglement of formation, distillable entanglement, relative entropy of entanglement, and squashed entanglement. By considering the distilled entanglement as a function of the success probability of the distillation protocol, we show that the logarithmic negativity surpasses the bound on deterministic entanglement distribution at a relatively large probability of success. This is in contrast to the other measures which would only be able to do so at much lower probabilities, hence demonstrating that logarithmic negativity alone is inadequate for assessing the performance of the distillation protocol. In addition to this result, we also observed an increase in the distillable entanglement by making use of upper and lower bounds to estimate this quantity. We thus demonstrate the utility of these theoretical tools in an experimental setting.
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.