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arXiv:1002.2511 · New Journal of Physics

Additivity and non-additivity of multipartite entanglement measures

Huangjun Zhu, Lin Chen, Masahito Hayashi

Published Aug 2, 2010• 46 citations
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Abstract

We study the additivity property of three multipartite entanglement measures, i.e. the geometric measure of entanglement (GM), the relative entropy of entanglement and the logarithmic global robustness. Firstly, we show the additivity of GM of multipartite states with real and non-negative entries in the computational basis. Many states of experimental and theoretical interest have this property, e.g. Bell diagonal states, maximally correlated generalized Bell diagonal states, generalized Dicke states, the Smolin state and the generalization of Dür's multipartite bound entangled states. We also prove the additivity of the other two measures for some of these examples. Secondly, we show the non-additivity of GM of all antisymmetric states of three or more parties. We also provide a unified explanation of the non-additivity of the three measures of the antisymmetric projector states. In addition, we derive analytical formulae for the three measures of one copy and two copies of the antisymmetric projector states, respectively. Thirdly, we show, with a statistical approach, that almost all multipartite pure states with a sufficiently large number of parties are nearly maximally entangled with respect to GM and relative entropy of entanglement, and they have non-additive GM. Hence, more states may be suitable for universal quantum computation if measurements can be performed on two copies of the resource states. We also show that almost all the multipartite pure states cannot be produced reversibly with the combination of multipartite GHZ states under asymptotic LOCC, unless the relative entropy of entanglement is non-additive for generic multipartite pure states.

DOI
10.1088/1367-2630/12/8/083002

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We study the additivity property of three multipartite entanglement measures, i.e. the geometric measure of entanglement (GM), the relative entropy of entanglement and the logarithmic global robustness. Firstly, we show the additivity of GM of multipartite states with real and non-negative entries in the computational basis. Many states of experimental and theoretical interest have this property, e.g. Bell diagonal states, maximally correlated generalized Bell diagonal states, generalized Dicke states, the Smolin state and the generalization of Dür's multipartite bound entangled states. We also prove the additivity of the other two measures for some of these examples. Secondly, we show the non-additivity of GM of all antisymmetric states of three or more parties. We also provide a unified explanation of the non-additivity of the three measures of the antisymmetric projector states. In addition, we derive analytical formulae for the three measures of one copy and two copies of the antisymmetric projector states, respectively. Thirdly, we show, with a statistical approach, that almost all multipartite pure states with a sufficiently large number of parties are nearly maximally entangled with respect to GM and relative entropy of entanglement, and they have non-additive GM. Hence, more states may be suitable for universal quantum computation if measurements can be performed on two copies of the resource states. We also show that almost all the multipartite pure states cannot be produced reversibly with the combination of multipartite GHZ states under asymptotic LOCC, unless the relative entropy of entanglement is non-additive for generic multipartite pure states.
Several inequivalent definitions of the geometric measure of entanglement (GM) have been introduced and studied in the past. Here we review several known and new definitions, with the qualifying criterion being that for pure states the measure is a linear or logarithmic function of the maximal fidelity with product states. The entanglement axioms and properties of the measures are studied, and qualitative and quantitative comparisons are made between all definitions. Streltsov et al. [New J. Phys. 12, 123004 (2010)] proved the equivalence of two linear definitions of GM, whereas we show that the corresponding logarithmic definitions are distinct. Certain classes of states such as ``maximally correlated states'' and isotropic states are particularly valuable for this analysis. A little-known GM definition is found to be the first one to be both normalized and weakly monotonous, thus being a prime candidate for future studies of multipartite entanglement. We also find that a large class of graph states, which includes all cluster states, have a ``universal'' closest separable state that minimizes the quantum relative entropy, the Bures distance, and the trace distance.
Abstract We study systematically resource measures of coherence and entanglement based on Rényi relative entropies, which include the logarithmic robustness of coherence, geometric coherence, and conventional relative entropy of coherence together with their entanglement analogues. First, we show that each Rényi relative entropy of coherence is equal to the corresponding Rényi relative entropy of entanglement for any maximally correlated state. By virtue of this observation, we establish a simple operational connection between entanglement measures and coherence measures based on Rényi relative entropies. We then prove that all these coherence measures, including the logarithmic robustness of coherence, are additive. Accordingly, all these entanglement measures are additive for maximally correlated states. In addition, we derive analytical formulas for Rényi relative entropies of entanglement of maximally correlated states and bipartite pure states, which reproduce a number of classic results on the relative entropy of entanglement and logarithmic robustness of entanglement in a unified framework. Several nontrivial bounds for Rényi relative entropies of coherence (entanglement) are further derived, which improve over results known previously. Moreover, we determine all states whose relative entropy of coherence is equal to the logarithmic robustness of coherence. As an application, we provide an upper bound for the exact coherence distillation rate, which is saturated for pure states.
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Bound entangled states are states that are entangled but from which no entanglement can be distilled if all parties are allowed only local operations and classical communication. However, in creating these states one needs nonzero entanglement resources to start with. Here, the entanglement of two distinct multipartite bound entangled states is determined analytically in terms of a geometric measure of entanglement and a related quantity. The results are compared with those for the negativity and the relative entropy of entanglement.
In this paper, we investigate the characterization of multipartite entanglement based on the unified (q,s)(q,s)(q,s)-entropy framework. For bipartite quantum states, we propose an entanglement measure Eq,sA∣B(ρ)E_{q,s}^{A|B}(ρ)Eq,sA∣B​(ρ) based on unified entropy and derive several analytical lower bounds for it using local orthonormal observables. We demonstrate with concrete examples that our lower bounds provide tighter estimates of quantum entanglement than several existing analytical bounds. For multipartite quantum states, based on unified entropy, we propose two measures Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) for quantifying kkk-nonseparability, and prove that these measures satisfy several desirable properties, such as faithfulness, local unitary invariance, and convexity. Moreover, we rigorously prove that when the parameters qqq and sss are in certain ranges, Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) also satisfy monotonicity and strong monotonicity. Furthermore, we establish the order relations of the measures Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) with respect to the parameters qqq and sss for fixed quantum states. In addition, for fixed qqq and sss, we also give their order relations for any two pure states.
We study the geometric measure of entanglement (GM) of pure symmetric states related to rank 1 positive-operator-valued measures (POVMs) and establish a general connection with quantum state estimation theory, especially the maximum likelihood principle. Based on this connection, we provide a method for computing the GM of these states and demonstrate its additivity property under certain conditions. In particular, we prove the additivity of the GM of pure symmetric multiqubit states whose Majorana points under Majorana representation are distributed within a half sphere, including all pure symmetric three-qubit states. We then introduce a family of symmetric states that are generated from mutually unbiased bases and derive an analytical formula for their GM. These states include Dicke states as special cases, which have already been realized in experiments. We also derive the GM of symmetric states generated from symmetric informationally complete POVMs (SIC POVMs) and use it to characterize all inequivalent SIC POVMs in three-dimensional Hilbert space that are covariant with respect to the Heisenberg-Weyl group. Finally, we describe an experimental scheme for creating the symmetric multiqubit states studied in this article and a possible scheme for measuring the permanence of the related Gram matrix.