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

Abstract

Recent years have witnessed a controversy over Heisenberg's famous error-disturbance relation. Here we resolve the conflict by way of an analysis of the possible conceptualizations of measurement error and disturbance in quantum mechanics. We discuss two approaches to adapting the classic notion of root-mean-square error to quantum measurements. One is based on the concept of 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.

Quantum Errors and Disturbances: Response to Busch, Lahti and Werner

“…The commutators [μ f ,x i ] and [p f ,p i ] are typically non-zero, so the O quantities are typically not generalizations of the corresponding classical quantities, as Busch and co-workers have stressed [24,49]. The O quantities do, however, impose bounds on the D and C quantities, and this gives them an indirect physical interpretation.…”

“…We focus on Busch, Lahti and Werner's (BLW's) criticisms [2,[23][24][25][26] of the operator approach [27][28][29][30][31] to the description of quantum errors and disturbances. Their criticisms raise some issues that are highly relevant to the above discussion and that need to be settled if we hope to make progress.…”

Quantum measurement and uncertainty relations in photon polarization

“…Also, some of them are defined based on the RMS of measurement outcomes, and others are defined based on information-theoretic quantities. There are active discussions and debates on this matter [3,5,6]. Here, we introduce a state-dependent definition of the measurement error given in the general theory of quantum instruments proposed by Ozawa [2].…”

“…Correspondingly, there are a number of proposals for the measurement uncertainty relations based on different definitions of error and disturbance. Busch et al [5,30,31] proposed a definition of measurement error based on the RMS distance of the distributions between the original (Â) and the measurement (M A ) observables. The state-independent error is defined by taking the supremum of the RMS distance with respect to all the input states.…”

“…However, surprisingly, no commonly agreed definitions have been established on the measurement error and disturbance, and thus on uncertainty relations in quantum measurements. Recent theoretical progress in quantum measurements and uncertainty relations has revealed the new aspects of these issues [2,3,4,5,6]. In addition, the concept of weak measurement and weak value [7] has attracted great attention.…”