Quantum Supermaps are Characterized by Locality

TL;DR

This paper introduces a new, category-theoretic characterization of quantum supermaps based on locality and compositional principles, extending the concept to broader operational frameworks.

Contribution

It provides a novel axiomatic and diagrammatic approach to quantum supermaps, generalizing them to monoidal categories and probabilistic theories.

Findings

Locally-applicable transformations correspond to deterministic quantum supermaps.

The characterization applies to complex supermaps like the quantum switch.

The approach works in general convex spaces of quantum channels.

Abstract

We provide a new characterisation of quantum supermaps in terms of an axiom that refers only to sequential and parallel composition. Consequently, we generalize quantum supermaps to arbitrary monoidal categories and operational probabilistic theories. We do so by providing a simple definition of locally-applicable transformation on a monoidal category. The definition can be rephrased in the language of category theory using the principle of naturality, and can be given an intuitive diagrammatic representation in terms of which all proofs are presented. In our main technical contribution, we use this diagrammatic representation to show that locally-applicable transformations on quantum channels are in one-to-one correspondence with deterministic quantum supermaps. This alternative characterization of quantum supermaps is proven to work for more general multiple-input supermaps such as the quantum switch and on arbitrary normal convex spaces of quantum channels such as those defined by satisfaction of signaling constraints.