Characterizing Incomparability in Quantum Information

TL;DR

This paper investigates the concept of incomparability in quantum spectra, introducing measures and classifications to understand when spectra cannot be ordered by majorization, which impacts quantum information processing.

Contribution

It introduces methods to quantify and classify incomparability between spectra, extending the majorization framework in quantum information theory.

Findings

Spectra isoentropic by generalized entropies are strongly incomparable.

Inversion rank is proposed as a new classification tool.

Majorization relations help measure the scale of incomparability.

Abstract

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.