In quantum physics, it is very important to give the theoretical lower and upper bounds for the geometric measure of entanglement of a multipartite pure state with nonnegative amplitudes. Existing literature shows that the theoretical bounds can be obtained by the bounds of \(Z_2\) -spectral radius of a nonnegative tensor. In this paper, a part of conclusions on Perron-Frobenius Theorem of the \(Z_2\) -eigenpairs for a nonnegative tensor are extended to its \(Z_p\) -eigenpairs, where p is any positive integer. Subsequently, an upper bound of any \(Z_p\) -eigenvalue of a tensor is derived. And then, a lower bound of the ratio of the largest and smallest components of a positive \(Z_p\) -eigenvector of an irreducible and nonnegative tensor is provided. Finally, two numerical examples are given to show the effectiveness of the obtained bounds in estimating the geometric measure of entanglement.
