Absolutely $k$-Incoherent Quantum States and Spectral Inequalities for Factor Width of a Matrix

TL;DR

This paper characterizes quantum states that are absolutely $k$-incoherent based solely on their eigenvalues, providing spectral inequalities and conditions related to factor width of matrices, advancing understanding in quantum resource theory.

Contribution

It introduces the concept of absolutely $k$-incoherent states and derives spectral inequalities and conditions for their identification based on eigenvalues.

Findings

Derived necessary and sufficient conditions for absolute $k$-incoherence.

Connected spectral properties with hyperbolicity cones and elementary symmetric polynomials.

Extended the understanding of quantum states' incoherence using spectral inequalities.

Abstract

We investigate the set of quantum states that can be shown to be $k$-incoherent based only on their eigenvalues (equivalently, we explore which Hermitian matrices can be shown to have small factor width based only on their eigenvalues). In analogy with the absolute separability problem in quantum resource theory, we call these states "absolutely $k$-incoherent", and we derive several necessary and sufficient conditions for membership in this set. We obtain many of our results by making use of recent results concerning hyperbolicity cones associated with elementary symmetric polynomials.