Entropic measures of joint uncertainty: effects of lack of majorization

TL;DR

This paper investigates how different entropic measures quantify joint uncertainty in quantum systems, revealing that the lack of majorization affects the uncertainty relations and that standard duality relations do not always apply.

Contribution

It introduces a detailed analysis of Rényi entropies for joint quantum observables and highlights the impact of the absence of majorization on uncertainty measures.

Findings

Rényi entropies vary with the choice of uncertainty measure.

Standard duality relations do not capture all behaviors of joint uncertainty.

Lack of majorization explains discrepancies in uncertainty relations.

Abstract

We compute R\'enyi entropies for the statistics of a noisy simultaneous observation of two complementary observables in two-dimensional quantum systems. The relative amount of uncertainty between two states depends on the uncertainty measure used. These results are not reproduced by a more standard duality relation. We show that these behaviors are consistent with the lack of majorization relation between the corresponding statistics.