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

Abstract

The geometric measure of entanglement for a symmetric pure state with nonnegative amplitudes has attracted much attention. On the other hand, the spectral theory of nonnegative tensors (hypermatrices) has been developed rapidly. In this paper, we show how the spectral theory of nonnegative tensors can be applied to the study of the geometric measure of entanglement for a pure state with nonnegative amplitudes. Especially, an elimination method for computing the geometric measure of entanglement for symmetric pure multipartite qubit or qutrit states with nonnegative amplitudes is given. For symmetric pure multipartite qudit states with nonnegative amplitudes, a numerical algorithm with randomization is presented and proven to be convergent. We show that for the geometric measure of entanglement for pure states with nonnegative amplitudes, the nonsymmetric ones can be converted to the symmetric ones.

21 pages

References in corpus (4)

• The geometric measure of entanglement for symmetric states
• Finding the Spectral Radius of a Nonnegative Tensor
• The Spectral Theory of Tensors (Rough Version)
• The Minimum Hartree Value for the Quantum Entanglement Problem

Cited by in corpus (7)

• Finding the Spectral Radius of a Nonnegative Tensor
• Hermitian tensor and quantum mixed state
• The Quantum Eigenvalue Problem and Z-Eigenvalues of Tensors
• Computing tensor eigenvalues via homotopy methods
• Geometric measure of entanglement of multipartite mixed states
• Sherman-Morrison-Woodbury Identity for Tensors
• Iterative methods for computing U-eigenvalues of non-symmetric complex tensors with application in quantum entanglement