Characterizing Incomparability in Quantum Information

The theory of majorization has seen substantial application in quantum information. Its framework predicates on the comparability between real vectors. We explore the antithesis of this premise, namely, incomparability. Specifically, we provide ways to measure the incomparability between a pair of spectra. We show that distinct spectra isoentropic by generalized entropies are mutually strongly incomparable. The inversion rank is proposed to classify incomparability. Majorization relations are advanced to probe the scale of incomparability, referred to as inconvertibility.