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

Error and disturbance are fundamental concepts in quantum measurements; therefore the question of how to formulate these two concepts has come to be considered so far. However, we still do not have the ultimate answer, hopefully unifying the existing definitions, to the question of what the error and disturbance are. Here we show that error and disturbance can be defined as special cases of the irreversibility of quantum processes. In our formulation, by applying a channel conversion---quantum comb in quantum information theory---to the measuring process, we convert its error and disturbance into the irreversibility of time evolution of an ancillary qubit system. The re-definitions provide fruitful byproducts: First, we can unify the existing definitions of error and disturbance as special aspects of irreversibility. Second, we extend the quantitative Wigner--Araki--Yanase theorem---a universal restriction on measurement implementation under a conservation law---to error and disturbance of arbitrary definitions. Third, we provide a novel treatment of out-of-time-orderd-correlator (OTOC)---a measure of quantum chaos in a quantum many-body system---as irreversibility, and its experimental evaluation method.