Incompatibility of observables, channels and instruments in information theories

TL;DR

This paper explores the concept of compatibility among tests in operational probabilistic theories, revealing conditions under which tests are incompatible and linking incompatibility to the fundamental disturbance of information extraction.

Contribution

It introduces and compares strong and weak notions of test compatibility, providing necessary and sufficient conditions for incompatibility in general theories.

Findings

Strong and weak compatibility coincide for observation tests.

Existence of weakly compatible but not strongly compatible channels.

Incompatibility arises when some information cannot be obtained without disturbance.

Abstract

Every theory of information, including classical and quantum, can be studied in the framework of operational probabilistic theories--where the notion of test generalizes that of quantum instrument, namely a collection of quantum operations summing to a channel, and simple rules are given for the composition of tests in parallel and in sequence. Here we study the notion of compatibility for tests of an operational probabilistic theory. Following the quantum literature, we first introduce the notion of strong compatibility, and then we illustrate its ultimate relaxation, that we deem weak compatibility. It is shown that the two notions coincide in the case of observation tests--which are the counterpart of quantum POVMs--while there exist weakly compatible channels that are not strongly compatible. We prove necessary and sufficient conditions for a theory to exhibit incompatible tests. We show that a theory admits of incompatible tests if and only if some information cannot be extracted without disturbance.