In this paper, we present some new \(Z\)-eigenvalue inclusion theorem for tensors by categorizing the entries of tensors, and prove that these sets are more precise than existing results. On this basis, some lower and upper bounds for the \(Z\)-spectral radius of weakly symmetric nonnegative tensors are proposed, which improves some of the existing results. As applications, we give some estimates of the best rank-one approximation rate in weakly symmetric nonnegative tensors and the maximal orthogonal rank of real orthogonal tensors, and our results are more precise than existing result in some situations. In particular, for a given symmetric multipartite pure state with nonnegative amplitudes in real field, some theoretical lower and upper bounds for the geometric measure of entanglement are also derived in terms of the bounds for \(Z\)-spectral radius. Numerical examples are given to illustrate validity and superiority of our results.

在本文中，我们通过对张量的项进行分类，提出了一些新的\（Z \）-特征值包含定理，并证明这些集合比现有结果更精确。在此基础上，\（Z \）的一些上下限提出了弱对称非负张量的谱半径，这改善了一些现有的结果。作为应用，我们给出了弱对称非负张量的最佳秩近似率和实正交张量的最大正交秩的估计值，在某些情况下，我们的结果比现有结果更为精确。特别地，对于给定的对称多部分纯态，其在实场中具有非负振幅，还根据\（Z \）-光谱半径的边界导出了一些理论上的纠缠几何下限和上限。数值例子说明了我们的结果的有效性和优越性。