Tensor eigenvalues and entanglement of symmetric states

TL;DR

This paper explores how tensor eigenvalues can reveal entanglement properties of symmetric quantum states, extending previous spectral-entanglement links and showing correlations for higher angular momentum states.

Contribution

It introduces the application of tensor eigenvalues to multipartite symmetric states and demonstrates their relation to entanglement, especially for higher angular momentum cases.

Findings

For spin-1 states, the smallest tensor eigenvalue's positivity indicates separability.

For higher angular momentum states, a correlation exists between entanglement and the smallest tensor eigenvalue.

The work extends spectral-entanglement connections beyond previous limitations.

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.