Universal Uncertainty Relations

TL;DR

This paper introduces a universal framework for quantum uncertainty relations based on Schur-concave functions and majorization, extending previous entropy-based approaches to encompass a broader class of uncertainty quantifiers.

Contribution

It establishes that Schur-concave functions are the most general uncertainty measures and develops a universal, majorization-based uncertainty relation that extends prior entropy-based results.

Findings

Introduces a universal uncertainty relation using majorization.

Extends previous entropy-based uncertainty relations.

Provides a family of scalar uncertainty measures from the vector relation.

Abstract

Uncertainty relations are a distinctive characteristic of quantum theory that impose intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs \emph{entropic measures} to quantify the lack of knowledge associated with measuring non-commuting observables. However, there is no fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation. Starting from a very reasonable assumption of invariance under mere relabelling of the measurement outcomes, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a fine-grained uncertainty relation that is given in terms of the majorization order between two probability vectors, \textcolor{black}{significantly extending a majorization-based uncertainty relation first introduced in [M. H. Partovi, Phys. Rev. A \textbf{84}, 052117 (2011)].} Such a vector-type uncertainty relation generates an infinite family of distinct scalar uncertainty relations via the application of arbitrary uncertainty quantifiers. Our relation is therefore universal and captures the essence of uncertainty in quantum theory.