Soundness and completeness of quantum root-mean-square errors

TL;DR

This paper introduces an improved, operationally definable quantum root-mean-square error that accurately characterizes measurement precision and preserves the validity of uncertainty relations, addressing issues with previous definitions.

Contribution

It proposes a new state-dependent quantum rms error that maintains universal uncertainty relations and aligns with experimental results, resolving prior conceptual issues.

Findings

The new error measure is operationally definable and state-dependent.

It preserves the validity of universally valid uncertainty relations.

Experimental confirmations support the new formulation.

Abstract

Defining and measuring the error of a measurement is one of the most fundamental activities in experimental science. However, quantum theory shows a peculiar difficulty in extending the classical notion of root-mean-square (rms) error to quantum measurements. A straightforward generalization based on the noise-operator was used to reformulate Heisenberg's uncertainty relation for the accuracy of simultaneous measurements to be universally valid and made the conventional formulation testable to observe its violation. Recently, its reliability was examined based on an anomaly that the error vanishes for some imprecise measurements, in which the meter does not commute with the measured observable. Here, we propose an improved definition for a quantum generalization of the classical rms error, which is state-dependent, operationally definable, and perfectly characterizes precise measurements. Moreover, it is shown that the new notion maintains the previously obtained universally valid uncertainty relations and their experimental confirmations without changing their forms and interpretations, in contrast to a prevailing view that a state-dependent formulation for measurement uncertainty relation is not tenable.