Incompatible measurements on quantum causal networks

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.

26 pages, 10 figures, published version

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Cited by in corpus (13)

• Quantum Steering
• Optimal quantum networks and one-shot entropies
• Incompatibility of quantum channels
• All measurements in a probabilistic theory are compatible if and only if the state space is a simplex
• Conditions on the existence of maximally incompatible two-outcome measurements in General Probabilistic Theory
• Incompatible measurements in a class of general probabilistic theories
• Quantification of quantum dynamics with input-output games
• A necessary condition for incompatibility of observables in general probabilistic theories
• Conditions for the compatibility of channels in general probabilistic theory and their connection to steering and Bell nonlocality
• Popescu-Rohrlich box implementation in general probabilistic theory of processes
• Classical and Quantum Causal Interventions
• Entropic Bounds For Unitary Testers and Mutually Unbiased Unitary Bases
• Structure of quantum and classical implementations of Popescu-Rohrlich box