Tensor eigenvalues and entanglement of symmetric states

Abstract

Tensor eigenvalues and eigenvectors have been introduced in the recent mathematical literature as a generalization of the usual matrix eigenvalues and eigenvectors. We apply this formalism to a tensor that describes a multipartite symmetric state or a spin state, and we investigate to what extent the corresponding tensor eigenvalues contain information about the multipartite entanglement (or, equivalently, the classicality) of the state. This extends previous results connecting entanglement to spectral properties related to the state. While for spin-1 states the positivity of the smallest tensor eigenvalue is equivalent to separability, we show that for higher values of the angular momentum there is a correlation between entanglement and the value of the smallest tensor eigenvalue.

9 pages, 5 figures

References in corpus (6)

• Entanglement and permutational symmetry
• The geometric measure of entanglement for symmetric states
• Multiqubit symmetric states with high geometric entanglement
• The maximally entangled symmetric state in terms of the geometric measure
• Characterizing multiparticle entanglement in symmetric N-qubit states via negativity of covariance matrices
• Tensor Representation of Spin States