Absolutely <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> -incoherent quantum states and spectral inequalities for the factor width of a matrix

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