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arXiv:1308.0806 · Physical Review A

Comparison of different definitions of the geometric measure of entanglement

Lin Chen, Martin Aulbach, Michal Hajdušek

Published Apr 7, 2014• 34 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.89.042305

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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.
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.
We exactly evaluate a number of multipartite entanglement measures for a class of graph states, including d -dimensional cluster states ( d = 1,2,3), the Greenberger–Horne–Zeilinger states, and some related mixed states. The entanglement measures that we consider are continuous, 'distance from separable states' measures, including the relative entropy, the so-called geometric measure, and robustness of entanglement. We also show that for our class of graph states these entanglement values give an operational interpretation as the maximal number of graph states distinguishable by local operations and classical communication (LOCC), as well as supplying a tight bound on the fixed letter classical capacity under LOCC decoding.
We introduce a class of generalized geometric measures of entanglement. For pure quantum states of N\mathit{N}N elementary subsystems, they are defined as the distances from the sets of K\mathit{K}K-separable states (K=2,…,N)(K=2,\dots{},N)(K=2,…,N). The entire set of generalized geometric measures provides a quantification and hierarchical ordering of the different bipartite and multipartite components of the global geometric entanglement, and allows discrimination among the different contributions. The extended measures are applied to the study of entanglement in different classes of N\mathit{N}N-qubit pure states. These classes include W\mathit{W}W and Greenberger-Horne-Zeilinger (GHZ) states, and their symmetric superpositions; symmetric multimagnon states; cluster states; and, finally, asymmetric generalized W\mathit{W}W-like superposition states. We discuss in detail a general method for the explicit evaluation of the multipartite components of geometric entanglement, and we show that the entire set of geometric measures establishes an ordering among the different types of bipartite and multipartite entanglement. In particular, it determines a consistent hierarchy between GHZ and W\mathit{W}W states, clarifying the original result of Wei and Goldbart that W\mathit{W}W states possess a larger global entanglement than GHZ states. Furthermore, we show that all multipartite components of geometric entanglement in symmetric states obey a property of self-similarity and scale invariance with the total number of qubits and the number of qubits per party.
An important problem in quantum information theory is the quantification of entanglement in multipartite mixed quantum states. In this work, a connection between the geometric measure of entanglement and a distance measure of entanglement is established. We present a new expression for the geometric measure of entanglement in terms of the maximal fidelity with a separable state. A direct application of this result provides a closed expression for the Bures measure of entanglementof two qubits. We also prove that the number of elements in an optimal decomposition w.r.t. the geometric measure of entanglement is bounded from above by the Caratheodory bound, and we find necessary conditions for the structure of an optimal decomposition.
We present a revised geometric measure of entanglement (RGME) which is just certain generalization of geometric measure of entanglement (GME). The revised version is an entanglement monotone. Some useful inequalities about RGME are deduced. For example, we give the formulae of RGME for the two-parameter class of states in a 2 ⊗ n quantum system, the two particles' high dimensional maximally entangled mixed state, the isotropic state including the n -particle d-level case and two multipartite bound entangled states. The results show there is a relation , and then RGME is an appropriate measure of entanglement.
The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement, already present in a number of settings [A. Shimony, Ann. NY. Acad. Sci. 755, 675 (1995); H. Barnum and N. Linden, J. Phys. A: Math. Gen. 34, 6787 (2001)], is explored for bipartite and multipartite pure and mixed states. The measure is determined analytically for arbitrary two-qubit mixed states and for generalized Werner and isotropic states, and is also applied to certain multipartite mixed states. In particular, a detailed analysis is given for arbitrary mixtures of three-qubit Greenberger-Horne-Zeilinger, W, and inverted-WWW states. Along the way, we point out connections of the geometric measure of entanglement with entanglement witnesses and with the Hartree approximation method.