Quantifying incompatibility beyond entropic uncertainty

TL;DR

This paper introduces new measures for quantum incompatibility that extend beyond traditional entropic uncertainty relations, capturing incompatibility even when observables commute over subspaces, and provides practical bounds and evaluations.

Contribution

It proposes two novel incompatibility measures that go beyond entropic uncertainty relations and offers methods to compute bounds for general sets of observables.

Findings

Incompatibility measures are non-zero even when observables commute over subspaces.

Both measures are bounded below by Tsallis entropy-based EURs.

Explicit evaluation of qubit observables demonstrates the measures' effectiveness.

Abstract

We study two operational approaches to quantifying incompatibility that depart significantly from the well known entropic uncertainty relation (EUR) formalism. Both approaches result in incompatibility measures that yield non-zero values even when the pair of incompatible observables commute over a subspace, unlike EURs which give a zero lower bound in such cases. Here, we explicitly show how these measures go beyond EURs in quantifying incompatibility: For any set of quantum observables, we show that both incompatibility measures are bounded from below by the corresponding EURs for the Tsallis ($T_{2}$) entropy. We explicitly evaluate the incompatibility of a pair of qubit observables in both operational scenarios. We also obtain an efficiently computable lower bound for the mutually incompatibility of a general set of observables.