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

Abstract: Since the proposal of Heisenberg's uncertainty principle, error and disturbance of quantum measurements have been fundamental notions in quantum physics. As is often the case when defining physical quantities in quantum physics, there is no single way to define these two notions, and many independent definitions of them have been given. Here, we establish a novel formulation defining the error and disturbance as special cases of the irreversibility in quantum processes. The formulation enables us to apply the knowledge of irreversibility in stochastic thermodynamics and quantum information theory to the error and disturbance in quantum measurements. To demonstrate this strength, we provide three byproducts: First, we unify the existing formulations of error and disturbance. Second, we extend the quantitative Wigner--Araki--Yanase theorem -- a universal restriction on measurement implementation under a conservation law -- to errors and disturbances of arbitrary definitions and processes. Third, we reveal that our formulation covers the out-of-time-orderd-correlator -- a measure of quantum chaos in a quantum many-body system -- as the irreversibility in analogy with the measurement context, and provide its experimental evaluation method.