SOME NEW Z-EIGENVALUE LOCALIZATION SETS FOR EVEN-ORDER TENSORS AND THEIR APPLICATION IN THE GEOMETRIC MEASURE OF ENTANGLEMENT

. We study Z-eigenvalue problems motivated by the geometric measure of quantum entanglement. Firstly, a new Z-eigenvalue localization set with parameters by Z-identity tensors is given. Secondly, some new Brauer-type Zeigenvalue inclusion sets by classifying the index set are presented. Thirdly, some lower and upper bounds for the Z-spectral radius of weakly symmetric nonnegative tensors are proposed, which improves some of the existing results. Finally, based on the connection between the geometric measure of entanglement of weakly symmetric pure states with nonnegative amplitudes and the Z-spectral theory of weakly symmetric nonnegative tensors, we propose sharp lower and upper bounds for the geometric measure of entanglement of multipartite pure states for two kinds of geometric measures with different definitions, respectively. The given numerical experiments are reported to show the efficiency of our results.