Characterizing and quantifying the incompatibility of quantum instruments

TL;DR

This paper investigates the incompatibility of quantum instruments, introducing a robustness measure, establishing bounds, and exploring the properties of post-processing and specific families of instruments.

Contribution

It introduces the incompatibility robustness for quantum instruments and analyzes their parallel compatibility, providing bounds and characterizing compatible families.

Findings

Derived universal bounds on incompatibility robustness.

Proved post-processing as a free operation for parallel compatibility.

Identified families of instruments with tight bounds and compatible indecomposable instruments.

Abstract

Incompatibility of quantum devices is one of the cornerstones of quantum theory, and the incompatibility of quantum measurements and channels has been linked to quantum advantage in certain information theoretic tasks. In this work, we focus on the less well-explored question of the incompatibility of quantum instruments, that is, devices that describe the measurement process in its entirety, accounting for both the classical measurement outcome and the quantum post-measurement state. In particular, we focus on the recently introduced notion of parallel compatibility of instruments, which has been argued to be a natural notion of instrument compatibility. We introduce -- similarly to the case of measurements and channels -- the incompatibility robustness of quantum instruments and derive universal bounds on it. We then prove that post-processing of quantum instruments is a free operation for parallel compatibility. Last, we provide families of instruments for which our bounds are tight, and families of compatible indecomposable instruments.