We study tensor analysis problems motivated by the geometric measure of quantum entanglement. We define 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. We 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. We convert the geometric measure of the entanglement problem to an algebraic equation system problem. A numerical example shows that a symmetric real tensor may have a best complex rankone approximation that is better than its best real rank-one approximation, which implies that the absolute-value largest Z-eigenvalue is not always the geometric measure of entanglement.
- Geometric measure of entanglement
- Symmetric real tensor
- The best rank-one approximation
- Unitary eigenvalue (U-eigenvalue)
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