How well can one jointly measure two incompatible observables on a given quantum state?

TL;DR

This paper investigates the fundamental limits of jointly measuring two incompatible quantum observables, providing a new tight relation that quantifies the optimal trade-off between measurement errors and disturbance.

Contribution

It introduces a novel, tight relation that characterizes the optimal error trade-off in approximate joint measurements of incompatible observables in quantum mechanics.

Findings

Derived a new, tight error trade-off relation for joint measurements.

Characterized the disturbance caused by approximate measurements.

Provided stronger error-disturbance relations for incompatible observables.

Abstract

Heisenberg's uncertainty principle is one of the main tenets of quantum theory. Nevertheless, and despite its fundamental importance for our understanding of quantum foundations, there has been some confusion in its interpretation: although Heisenberg's first argument was that the measurement of one observable on a quantum state necessarily disturbs another incompatible observable, standard uncertainty relations typically bound the indeterminacy of the outcomes when either one or the other observable is measured. In this paper, we quantify precisely Heisenberg's intuition. Even if two incompatible observables cannot be measured together, one can still approximate their joint measurement, at the price of introducing some errors with respect to the ideal measurement of each of them. We present a new, tight relation characterizing the optimal trade-off between the error on one observable versus the error on the other. As a particular case, our approach allows us to characterize the disturbance of an observable induced by the approximate measurement of another one; we also derive a stronger error-disturbance relation for this scenario.