Uncertainty relations: An operational approach to the error-disturbance tradeoff

TL;DR

This paper introduces an operational framework for quantum error and disturbance, deriving new uncertainty relations that are applicable to finite-dimensional systems and position-momentum, with implications for quantum information processing.

Contribution

It proposes an operational definition of error and disturbance based on distinguishability, leading to new Heisenberg-type uncertainty relations that avoid conceptual issues of traditional measures.

Findings

Derived uncertainty relations for joint measurability and error-disturbance

Applicable to finite-dimensional systems and position-momentum

Connected wave-particle duality to error-disturbance relations

Abstract

The notions of error and disturbance appearing in quantum uncertainty relations are often quantified by the discrepancy of a physical quantity from its ideal value. However, these real and ideal values are not the outcomes of simultaneous measurements, and comparing the values of unmeasured observables is not necessarily meaningful according to quantum theory. To overcome these conceptual difficulties, we take a different approach and define error and disturbance in an operational manner. In particular, we formulate both in terms of the probability that one can successfully distinguish the actual measurement device from the relevant hypothetical ideal by any experimental test whatsoever. This definition itself does not rely on the formalism of quantum theory, avoiding many of the conceptual difficulties of usual definitions. We then derive new Heisenberg-type uncertainty relations for both joint measurability and the error-disturbance tradeoff for arbitrary observables of finite-dimensional systems, as well as for the case of position and momentum. Our relations may be directly applied in information processing settings, for example to infer that devices which can faithfully transmit information regarding one observable do not leak any information about conjugate observables to the environment. We also show that Englert's wave-particle duality relation [PRL 77, 2154 (1996)] can be viewed as an error-disturbance uncertainty relation.