Error and Disturbance as Irreversibility with Applications: Unified Definition, Wigner--Araki--Yanase Theorem and Out-of-Time-Order Correlator

Defining an error of measurement has long been a foundational problem in science: even in classical experiments, data are statistical and admit no single universally optimal definition of error. In quantum mechanics, the challenge deepens: observed entities often lack preexisting definite values, and the act of measurement unavoidably disturbs the system of interest. Consequently, both error and disturbance must be quantified, and various definitions have been proposed to date. However, a unified perspective for understanding the differences and similarities among these diverse definitions of error and disturbance, and an operational framework for distinguishing between them, remain elusive. In this Letter, we propose a novel framework for defining error and disturbance using irreversibility. Our framework converts the error and disturbance of a quantum measurement of a system under consideration into the irreversibility of an ancillary qubit system, using a quantum comb composed of loss and recovery processes. The mechanism enables us to make the operational distinction that error uses the classical outputs, while disturbance uses the quantum outputs of the measurement in the recovery process. Furthermore, our framework yields several key consequences: (i) it encompasses existing definitions, (ii) it establishes a universal constraint on error and disturbance defined by any measure of an arbitrary quantum process under a conservation law, and (iii) it reveals an operational connection between irreversibility and the out-of-time-ordered correlator (OTOC), a metric of quantum chaos. It also provides a constraint on the OTOC under a conservation law and a method for its experimental evaluation, which is demonstrated on a quantum processor.