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Measures of entanglement

Vlatko Vedral

Published Sep 28, 2006• 4 citations
Publisher

Abstract

Abstract This chapter discusses how entangled two states are. One obvious way is to consider the distillation procedure, in which the entanglement of any state is quantified by how many fully entangled qubits can be distilled from an entangled state. However, it is not easy to see how to distil singlets out of n copies of a general mixed state. Hence, it is necessary to first see what can be done with pure states. This chapter discusses the distillation of multiple copies of a pure state, analogy between quantum entanglement and the Carnot cycle, properties of entanglement measures, entanglement of pure states, entanglement of mixed states, measures of entanglement derived from relative entropy, relationship between classical information and entanglement, and links between entanglement and thermodynamics.

DOI
10.1093/acprof:oso/9780199215706.003.0010

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Abstract This chapter discusses how entangled two states are. One obvious way is to consider the distillation procedure, in which the entanglement of any state is quantified by how many fully entangled qubits can be distilled from an entangled state. However, it is not easy to see how to distil singlets out of n copies of a general mixed state. Hence, it is necessary to first see what can be done with pure states. This chapter discusses the distillation of multiple copies of a pure state, analogy between quantum entanglement and the Carnot cycle, properties of entanglement measures, entanglement of pure states, entanglement of mixed states, measures of entanglement derived from relative entropy, relationship between classical information and entanglement, and links between entanglement and thermodynamics.
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
Quantum entanglement of pure states is usually quantified via the entanglement entropy, the von Neumann entropy of the reduced state. Entanglement entropy is closely related to entanglement distillation, a process for converting quantum states into singlets, which can then be used for various quantum technological tasks. The relation between entanglement entropy and entanglement distillation has been known only for the asymptotic setting, and the meaning of entanglement entropy in the single-copy regime has so far remained open. Here we close this gap by considering entanglement catalysis. We prove that entanglement entropy completely characterizes state transformations in the presence of entangled catalysts. Our results imply that entanglement entropy quantifies the amount of entanglement available in a bipartite pure state to be used for quantum information processing, giving asymptotic results an operational meaning also in the single-copy setup.
Distillation and purification of entanglement is an important field of quantum information both in theory and in experiment. This paper presents some main conclusions in this field, and discusses mainly the distillation of entanglement from finite copies of a mixed state. By using the conceptions of quasiseparable state and distillable subspace, we show a necessary condition and a sufficient and necessary condition of the distillability from finite copies of a mixed state. We also get the most efficient protocols of entanglement distillation for the case of finite copies. Finally we consider the generalization of these results to the case of infinite copies of a mixed state.
We study entanglement distillation and dilution of states and channels in the single-shot regime. With the help of a recently introduced conversion distance, we provide compact closed-form expressions for the dilution and distillation of pure states and show how this can be used to efficiently calculate these quantities on multiple copies of pure states. These closed-form expressions also allow us to obtain second-order asymptotics. We then prove that the ε-single-shot entanglement cost of mixed states is given exactly in terms of an expression containing a suitably smoothed version of the conditional max-entropy. For pure states, this expression reduces to the smoothed max-entropy of the reduced state, for which we provide a closed-form expression. Analogously, we provide a closed-form expression for the smoothed min-entropy and connect it to the ε-single-shot distillable entanglement. Based on these results, we bound the single-shot entanglement cost of channels. We then turn to the one-way entanglement distillation of states and channels and provide bounds in terms of a quantity we denote coherent information of entanglement.
As two of the most important entanglement measures--the entanglement of formation and the entanglement of distillation--have so far been limited to bipartite settings, the study of other entanglement measures for multipartite systems appears necessary. Here, connections between two other entanglement measures--the relative entropy of entanglement and the geometric measure of entanglement--are investigated. It is found that for arbitrary pure states the latter gives rise to a lower bound on the former. For certain pure states, some bipartite and some multipartite, this lower bound is saturated, and thus their relative entropy of entanglement can be found analytically in terms of their known geometric measure of entanglement. For certain mixed states, upper bounds on the relative entropy of entanglement are also established. Numerical evidence strongly suggests that these upper bounds are tight, i.e., they are actually the relative entropy of entanglement.
As two of the most important entanglement measures---the entanglement of formation and the entanglement of distillation---have so far been limited to bipartite settings, the study of other entanglement measures for multipartite systems appears necessary. Here, connections between two other entanglement measures---the relative entropy of entanglement and the geometric measure of entanglement---are investigated. It is found that for arbitrary pure states the latter gives rise to a lower bound on the former. For certain pure states, some bipartite and some multipartite, this lower bound is saturated, and thus their relative entropy of entanglement can be found analytically in terms of their known geometric measure of entanglement. For certain mixed states, upper bounds on the relative entropy of entanglement are also established. Numerical evidence strongly suggests that these upper bounds are tight, i.e., they are actually the relative entropy of entanglement.