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arXiv:1002.3049 · Physical Review A

Duality and the geometric measure of entanglement of general multiqubit W states

Sayatnova Tamaryan, Anthony Sudbery, Levon Tamaryan

Published May 17, 2010• 26 citations
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Abstract

We find the nearest product states for arbitrary generalized $W$ states of $n$ qubits, and show that the nearest product state is essentially unique if the $W$ state is highly entangled. It is specified by a unit vector in Euclidean $n$-dimensional space. We use this duality between unit vectors and highly entangled $W$ states to find the geometric measure of entanglement of such states.

DOI
10.1103/physreva.81.052319

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We show that when $N\ensuremath{\gg}1$ 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.
We solve stationarity equations of the geometric measure of entanglement for multi-qubit W-type states. In this way we compute analytically the maximal overlap of one-parameter nnn-qubit and two-parameter four-qubit W-type states and their nearest product states. Possible extensions to arbitrary W-type states and geometrical interpretations of these results are discussed in detail.
The geometric measure of entanglement is the distance or angle between an entangled target state and the nearest unentangled state. Often one considers the geometric measure of entanglement for highly symmetric entangled states because it simplifies the calculations and allows for analytic solutions. Although some symmetry is required in order to deal with large numbers of qubits, we are able to loosen significantly the restrictions on the highly symmetric states considered previously, and consider several generalizations of the coefficients of both target and unentangled states. This allows us to compute the geometric entanglement measure for larger and more relevant classes of states.
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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.