Colloquium : Quantum root-mean-square error and measurement uncertainty relations

Heisenberg's uncertainty principle is one of the pillars of quantum mechanics. In this Colloquium, issues arising with the use of the noise-operator concept for quantifying measurement errors are analyzed. An alternative way of adapting the classical concept of root-mean-square error to quantum measurements is presented, leading to Heisenberg-type measurement uncertainty relations. Recent years have witnessed a controversy over Heisenberg's famous error-disturbance relation. Here the conflict is resolved by way of an analysis of the possible conceptualizations of measurement error and disturbance in quantum mechanics. Two approaches to adapting the classic notion of root-mean-square error to quantum measurements are discussed. One is based on the concept of a noise operator; its natural operational content is that of a mean deviation of the values of two observables measured jointly, and thus its applicability is limited to cases where such joint measurements are available. The second error measure quantifies the differences between two probability distributions obtained in separate runs of measurements and is of unrestricted applicability. We show that there are no nontrivial unconditional joint-measurement bounds for state-dependent errors in the conceptual framework discussed here, while Heisenberg-type measurement uncertainty relations for state-independent errors have been proven.