Geometric measure of entanglement and U-eigenvalues of tensors (Q2877079)

The paper studies tensor analysis problems motivated by the geometric measure of quantum entanglement.NEWLINENEWLINEThe first section is an introduction in the nature of the subject.NEWLINENEWLINEIn the second section, reviewing the geometric measure of the entanglement problem, the authors introduce the concept of the unitary eigenvalue (U-eigenvalue) of a complex tensor, the unitary symmetric eigenvalue (US-eigenvalue) of a symmetric complex tensor and the best complex rank-one approximation.NEWLINENEWLINEIn the third section, the authors obtain an upper bound on the number of distinct US-eigenvalues of symmetric tensors and count all US-eigenpairs with nonzero eigenvalues of symmetric tensors.NEWLINENEWLINEIn the fourth section, they convert the geometric measure of the entanglement problem to an algebraic equation system problem. A numerical example is also analysed: the computation of US-eigenpairs of two 3rd-order 2-dimensional symmetric real tensors. The results show that some symmetric real tensors have better complex rank-one approximations than real rank-one approximations, which implies that not all the absolute-value largest Z-eigenvalues of symmetric real tensors are the geometric measures of symmetric pure states.NEWLINENEWLINEThe main conclusions are within the last section.