Spectral Properties of Symmetric Quantum States and Symmetric Entanglement Witnesses

TL;DR

This paper studies the spectral characteristics of symmetric quantum states and entanglement witnesses, addressing inverse eigenvalue problems and separability preservation under symmetric unitaries, with insights contrasting non-symmetric cases.

Contribution

It introduces the inverse eigenvalue problem for symmetric entanglement witnesses and characterizes symmetric states that remain separable under all symmetric unitaries, filling gaps in symmetric quantum entanglement theory.

Findings

Spectra of symmetric entanglement witnesses have specific constraints.

Certain symmetric states remain separable under all symmetric unitaries.

Contrasts with non-symmetric spectral and separability properties.

Abstract

We introduce and explore two questions concerning spectra of operators that are of interest in the theory of entanglement in symmetric (i.e., bosonic) quantum systems. First, we investigate the inverse eigenvalue problem for symmetric entanglement witnesses -- that is, we investigate what their possible spectra are. Second, we investigate the problem of characterizing which separable symmetric quantum states remain separable after conjugation by an arbitrary unitary acting on symmetric space -- that is, which states are separable in every orthonormal symmetric basis. Both of these questions have been investigated thoroughly in the non-symmetric setting, and we contrast the answers that we find with their non-symmetric counterparts.