Incompatible measurements on quantum causal networks

TL;DR

This paper extends the concept of quantum incompatibility to measurements testing dynamical processes, characterizes their incompatibility conditions, and introduces a robustness measure, with applications to quantum process testing.

Contribution

It provides necessary and sufficient conditions for incompatibility of quantum testers and introduces a quantitative robustness measure.

Findings

Robustness is bounded by the distinguishability of interaction sequences.

Maximum robustness occurs with perfectly distinguishable interactions.

Theoretical results are exemplified with single-photon polarization process testing.

Abstract

The existence of incompatible measurements, epitomized by Heisenberg's uncertainty principle, is one of the distinctive features of quantum theory. So far, quantum incompatibility has been studied for measurements that test the preparation of physical systems. Here we extend the notion to measurements that test dynamical processes, possibly consisting of multiple time steps. Such measurements are known as testers and are implemented by interacting with the tested process through a sequence of state preparations, interactions, and measurements. Our first result is a characterization of the incompatibility of quantum testers, for which we provide necessary and sufficient conditions. Then, we propose a quantitative measure of incompatibility. We call this measure the robustness of incompatibility and define it as the minimum amount of noise that has to be added to a set of testers in order to make them compatible. We show that (i) the robustness is lower bounded by the distinguishability of the sequence of interactions used by the tester and (ii) maximum robustness is attained when the interactions are perfectly distinguishable. The general results are illustrated in the concrete example of binary testers probing the time-evolution of a single-photon polarization.