# Witnessing incompatibility of quantum channels

## TL;DR

This paper introduces incompatibility witnesses for quantum channels, extending previous measurement incompatibility concepts, and shows they can be realized through state discrimination tasks, revealing advantages of incompatible channels.

## Contribution

It defines incompatibility witnesses for quantum channels and demonstrates their implementation via state discrimination, broadening the understanding of quantum incompatibility.

## Key findings

- Incompatibility witnesses can be realized as state discrimination tasks.
- Incompatible channels provide advantages in certain state discrimination scenarios.
- The framework applies to measurement-measurement, measurement-channel, and channel-channel pairs.

## Abstract

We introduce the notion of incompatibility witness for quantum channels, defined as an affine functional that is non-negative on all pairs of compatible channels and strictly negative on some incompatible pair. This notion extends the recent definition of incompatibility witnesses for quantum measurements. We utilize the general framework of channels acting on arbitrary finite dimensional von Neumann algebras, thus allowing us to investigate incompatibility witnesses on measurement-measurement, measurement-channel and channel-channel pairs. We prove that any incompatibility witness can be implemented as a state discrimination task in which some intermediate classical information is obtained before completing the task. This implies that any incompatible pair of channels gives an advantage over compatible pairs in some such state discrimination task.