Uncertainty relations from graph theory

Abstract

Quantum measurements are inherently probabilistic and quantum theory often forbids precisely predicting 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. For arbitrary observables, we obtain an upper bound on the sum of the square of their expectation values. We furthermore show how this bound behaves when the observables are imprecisely calibrated. As applications, our results can be straightforwardly used to formulate entropic uncertainty relations, separability criteria, and entanglement witnesses.