Quantum Supermaps are Characterized by Locality
Matt Wilson, Giulio Chiribella, Aleks Kissinger
TL;DR
The paper develops a locality-based axiom for quantum higher-order transformations, showing that quantum supermaps can be characterised purely by sequential/parallel composition and locality. It proves a one-to-one correspondence between locally-applicable transformations on quantum channels and standard quantum supermaps, and extends the framework to constrained spaces, multi-input cases, and arbitrary operational probabilistic theories. By recasting locality as naturality and using extension/dilation concepts, the authors provide a category-theoretic, diagrammatic foundation that unifies quantum and classical supermaps and opens paths to infinite-dimensional and OPT generalisations with potential implications for quantum gravity and causal structure. The work yields a principled, compositional basis for higher-order quantum processes that transcends reliance on compact closure or specific Hilbert-space formalisms, enabling broader applicability across theories of quantum information and beyond.
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.