Z-Eigenvalue Localization Sets for Tensors and the Applications in Rank-One Approximation and Quantum Entanglement

In this paper, we propose two Z-eigenvalue inclusive sets of tensors, and prove that our new inclusion sets are more precise than some existing results. Using the derived inclusion sets, we present new upper and lower bounds of the spectral radius of nonnegative weakly symmetric tensors. Further, we offer two applications of the obtained upper and lower bounds. One application is the best rank-one approximation rate. The other application is the geometric measure of quantum pure state entanglement with nonnegative amplitudes. Finally, numerical examples are given to illustrate the validity of the derived results.