The geometric measure of entanglement of multipartite states and the Z-eigenvalue of tensors

It is not easy to compute the entanglement of multipartite pure or mixed states, because it usually involves complex optimization. In this paper, we are devoted to the geometric measure of entanglement of multipartite pure or mixed state by the means of real tensor spectrum theory. On the basis of Z-eigenvalue inclusion theorem and the estimation of weakly symmetric nonnegative tensor Z-spectrum radius, we propose some theoretical upper and lower bounds of the geometric measure of entanglement for weakly symmetric pure state with nonnegative amplitudes for two kinds of geometric measures with different definitions, respectively. In addition, the upper bound of the geometric measure of entanglement is also applied to multipartite mixed state case.

Automated summary

In this paper, a method for computing the entanglement of multipartite pure and mixed states is proposed using real tensor spectrum theory. Theoretical upper and lower bounds for the geometric measure of entanglement are derived for weakly symmetric pure states, and the upper bound is also extended to multipartite mixed states.