Z-eigenvalue inclusion theorem of tensors and the geometric measure of entanglement of multipartite pure states

In our paper, we concentrate on the Z-eigenvalue inclusion theorem and its application in the geometric measure of entanglement of multipartite pure states. We present a new Z-eigenvalue inclusion theorem by virtue of the division and classification of tensor elements, and tighter bounds of Z-spectral radius of weakly symmetric nonnegative tensors are obtained. As applications, we present some theoretical upper and lower bounds of entanglement for symmetric pure state with nonnegative amplitudes for two kinds of geometric measures with different definitions, respectively.