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arXiv:1006.2432 · Physical Review A

Universal behavior of the geometric entanglement measure of many-qubit W states

Levon Tamaryan, Zaruhi Ohanyan, Sayatnova Tamaryan

Published Aug 10, 2010• 4 citations
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Abstract

We show that when $N\ensuremath{\gg}1$ the geometric entanglement measure of general $N$-qubit $W$ states, except maximally entangled $W$ states, is a one-variable function and depends only on the Bloch vector with the minimal $z$ component. Hence one can prepare a $W$ state with the required maximal product overlap by altering the Bloch vector of a single qubit. Next we compute analytically the geometric measure of large-scale $W$ states by describing these systems in terms of very few parameters. The final formula relates two quantities, namely the maximal product overlap and the Bloch vector, that can be easily estimated in experiments.

DOI
10.1103/physreva.82.022309

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We show that when N\ensuremath≫1N\ensuremath{\gg}1N\ensuremath≫1 the geometric entanglement measure of general NNN-qubit WWW states, except maximally entangled WWW states, is a one-variable function and depends only on the Bloch vector with the minimal zzz component. Hence one can prepare a WWW state with the required maximal product overlap by altering the Bloch vector of a single qubit. Next we compute analytically the geometric measure of large-scale WWW states by describing these systems in terms of very few parameters. The final formula relates two quantities, namely the maximal product overlap and the Bloch vector, that can be easily estimated in experiments.
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We provide methods for computing the geometric measure of entanglement for two families of pure states with both experimental and theoretical interests: symmetric multiqubit states with non-negative amplitudes in the Dicke basis and symmetric three-qubit states. In addition, we study the geometric measure of pure three-qubit states systematically in virtue of a canonical form of their two-qubit reduced states and derive analytical formulas for a three-parameter family of three-qubit states. Based on this result, we further show that the W state is the maximally entangled three-qubit state with respect to the geometric measure.
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We find the nearest product states for arbitrary generalized WWW states of nnn qubits, and show that the nearest product state is essentially unique if the WWW state is highly entangled. It is specified by a unit vector in Euclidean nnn-dimensional space. We use this duality between unit vectors and highly entangled WWW states to find the geometric measure of entanglement of such states.
Any pure three-qubit state is uniquely characterized by one phase and four positive parameters. The geometric measure of entanglement as a function of state parameters can have different expressions. Each of expressions has its own applicable domain and thus the whole state parameter space is divided into subspaces that are ranges of definition for corresponding expressions. The purpose of this paper is to examine the applicable domains for the most general qubit-interchange symmetric three-qubit states. First, we compute the eigenvalues of the non-linear eigenvalue equations and the nearest separable states for the permutation invariant three-qubit states with a fixed phase. Next, we compute the geometric entanglement measure, deduce the boundaries of all subspaces, and find allocations of highly and slightly entangled states. It is shown that there are three applicable domains when the phase factor is π/2π/2π/2 while other cases have only two domains. The emergence of the three domains is due to the appearance of the additional W-state. We show that most of highly entangled states reside near the boundaries of the domains and states located far from the boundaries become less-entangled and eventually go to the product states. The neighbors of W-state are generally more entangled than the neighbors of Greenberger-Horne-Zeilinger(GHZ) state from the aspect of the geometric measure. However, the range of the GHZ-neighbors is much more wider than the range of the W-neighbors.
In quantum physics, multiparticle systems are described by quantum states acting on tensor products of Hilbert spaces. This product structure leads to the distinction between product states and entangled states; moreover, one can quantify entanglement by considering the distance of a quantum state to the set of product states. The underlying optimization problem occurs frequently in physics and beyond, for instance in the computation of the injective tensor norm in multilinear algebra. Here, we introduce a method to determine the maximal overlap of a pure multiparticle quantum state with product states based on considering several copies of the pure state. This leads to three types of hierarchical approximations to the problem, all of which we prove to converge to the actual value. Besides allowing for the computation of the geometric measure of entanglement, our results can be used to tackle optimizations over stochastic local transformations, to find entanglement witnesses for weakly entangled bipartite states, and to design strong separability tests for mixed multiparticle states. Finally, our approach sheds light on the complexity of separability tests.