Uncertainty relations from graph theory

TL;DR

This paper introduces a novel approach to quantum uncertainty relations by leveraging graph theory, providing tight bounds for dichotomic observables and applications in entropic relations, separability, and entanglement detection.

Contribution

It establishes a new connection between quantum observables and graph theory to derive uncertainty relations applicable to various quantum measurement scenarios.

Findings

Derived tight uncertainty relations linked to maximum clique size in graphs

Applicable to entropic uncertainty relations and entanglement criteria

Provides a graph-theoretic framework for quantum measurement analysis

Abstract

Quantum measurements are inherently probabilistic and quantum theory often forbids to precisely predict the outcomes of simultaneous measurements. This phenomenon is captured and quantified through uncertainty relations. Although studied since the inception of quantum theory, the problem of determining the possible expectation values of a collection of quantum measurements remains, in general, unsolved. By constructing a close connection between observables and graph theory, we derive uncertainty relations valid for any set of dichotomic observables. These relations are, in many cases, tight, and related to the size of the maximum clique of the associated graph. As applications, our results can be straightforwardly used to formulate entropic uncertainty relations, separability criteria and entanglement witnesses.