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The geometric measure of entanglement for a symmetric pure state withnonnegative amplitudes has attracted much attention. On the other hand, thespectral theory of nonnegative tensors (hypermatrices) has been developedrapidly. In this paper, we show how the spectral theory of nonnegative tensorscan be applied to the study of the geometric measure of entanglement for a purestate with nonnegative amplitudes. Especially, an elimination method forcomputing the geometric measure of entanglement for symmetric pure multipartitequbit or qutrit states with nonnegative amplitudes is given. For symmetric puremultipartite qudit states with nonnegative amplitudes, a numerical algorithmwith randomization is presented and proven to be convergent. We show that forthe geometric measure of entanglement for pure states with nonnegativeamplitudes, the nonsymmetric ones can be converted to the symmetric ones.

The geometric measure of entanglement of pure states with nonnegative amplitudes and the spectral theory of nonnegative tensors