On the compatibility of quantum instruments

TL;DR

This paper investigates the compatibility of quantum instruments, introduces a new notion called parallel compatibility, and demonstrates it as a more complete framework than traditional compatibility for understanding quantum device interactions.

Contribution

The paper introduces parallel compatibility as a novel, complete concept for quantum instrument compatibility, surpassing the traditional notion.

Findings

Parallel compatibility captures measurement and channel compatibility.

Traditional compatibility is shown to be incomplete.

Parallel compatibility is proposed as the comprehensive definition.

Abstract

Incompatibility of quantum devices is a useful resource in various quantum information theoretical tasks, and it is at the heart of some fundamental features of quantum theory. While the incompatibility of measurements and quantum channels is well-studied, the incompatibility of quantum instruments has not been explored in much detail. In this work, we revise a notion of instrument compatibility introduced in the literature that we call traditional compatibility. Then, we introduce the new notion of parallel compatibility, and show that these two notions are inequivalent. Then, we argue that the notion of traditional compatibility is incomplete, and prove that while parallel compatibility captures measurement and channel compatibility, traditional compatibility does not. Hence, we propose parallel compatibility as the conceptually complete definition of compatibility of quantum instruments.