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International Journal of Software and Informatics

Geometric measure of quantum entanglement for multipartite mixed states

Shenglong Hu, Liqun Qi, Yisheng Song, Guofeng Zhang

Published Jan 1, 2014• 3 citations
OpenAlex

Abstract

The geometric measure of quantum entanglement of a pure state, defined by its distance to the set of pure separable states, is extended to multipartite mixed states. We characterize the nearest disentangled mixed state to a given mixed state with respect to the geometric measure by means of a system of equations. The entanglement eigenvalue for a mixed state is introduced. And we show that, for a given mixed state, its nearest disentangled mixed state is associated with its entanglement eigenvalue. Two numerical examples are used to demonstrate the effectiveness of the proposed method.

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The geometric measure of quantum entanglement of a pure state, defined by its distance to the set of pure separable states, is extended to multipartite mixed states. We characterize the nearest disentangled mixed state to a given mixed state with respect to the geometric measure by means of a system of equations. The entanglement eigenvalue for a mixed state is introduced. And we show that, for a given mixed state, its nearest disentangled mixed state is associated with its entanglement eigenvalue. Two numerical examples are used to demonstrate the effectiveness of the proposed method.
The geometric measure of entanglement for pure states has attracted much attention. On the other hand, the spectral theory of non-negative tensors (hypermatrices) has been developed rapidly. In this paper, we show how the spectral theory of non-negative tensors can be applied to the study of the geometric measure of entanglement for pure states. For symmetric pure multipartite qubit or qutrit states an elimination method is given. For symmetric pure multipartite qudit states, a numerical algorithm with randomization is presented. We also show that a nonsymmetric pure state can be augmented to a symmetric one whose amplitudes can be encoded in a non-negative symmetric tensor, so the geometric measure of entanglement can be calculated. Several examples, such as mGHZm\mathrm{GHZ}mGHZ states, WWW states, inverted WWW states, qudits, and nonsymmetric states, are used to demonstrate the power of the proposed methods. Given a pure state, one can always find a change of basis (a unitary transformation) so that all the probability amplitudes of the pure state are non-negative under the new basis. Therefore, the methods proposed here can be applied to a very wide class of multipartite pure states.
This study focuses on tensor Z-eigenvalue localization and its application in the geometric measure of entanglement for multipartite quantum states. A new Z-eigenvalue localization theorem and the bounds for the Z-spectral radius are derived, which are more precise than some of the existing results. On the other hand, we present theoretical bounds of the geometric measure of entanglement for a weakly symmetric multipartite quantum state with non-negative amplitudes by virtue of different distance measures. Numerical examples show that these conclusions are superior to the existing results in quantum physics in some cases.