# Necessary and Sufficient Conditions on Measurements of Quantum Channels

## TL;DR

This paper characterizes the conditions under which quantum supermaps can be realized as measurements on quantum channels, revealing fundamental limitations related to causality in quantum theory.

## Contribution

It provides necessary and sufficient conditions for implementing quantum supermaps as measurements, highlighting the role of causality violations.

## Key findings

- Not all supermaps transforming channels to CPTNI maps are physically realizable.
- Supermaps implementable as measurements are exactly those preserving CPTNI status under tensoring.
- The results connect the realizability of supermaps to causality principles in quantum theory.

## Abstract

Quantum supermaps are a higher-order generalization of quantum maps, taking quantum maps to quantum maps. It is known that any completely positive, trace non-increasing (CPTNI) map can be performed as part of a quantum measurement. By providing an explicit counterexample we show that, instead, not every quantum supermap sending a quantum channel to a CPTNI map can be realized in a measurement on quantum channels. We find that the supermaps that can be implemented in this way are exactly those transforming quantum channels into CPTNI maps even when tensored with the identity supermap. We link this result to the fact that the principle of causality fails in the theory of quantum supermaps.