Compatibility of quantum instruments

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 notion of parallel compatibility and show that these two notions are inequivalent. We then argue that the notion of traditional compatibility is conceptually incomplete and prove that, while parallel compatibility captures measurement and channel compatibility, traditional compatibility does not capture channel compatibility. Hence, we propose parallel compatibility as the conceptually complete definition of the compatibility of quantum instruments.

Automated summary

This paper investigates the compatibility of quantum instruments. By revising the concept of traditional compatibility and introducing the notion of parallel compatibility, it is shown that these two concepts are distinct. It is argued that traditional compatibility is conceptually incomplete and parallel compatibility is proposed as a more complete definition of the compatibility of quantum instruments.