Quantum Supermaps are Characterized by Locality

19 May 2022  ·  Matt Wilson, Giulio Chiribella, Aleks Kissinger ·

We prove that the sequential and parallel composition rules for quantum channels are all that need to be referenced to axiomatize quantum supermaps. To do so we provide a simple definition of locally-applicable transformation, which can be stated for arbitrary symmetric monoidal categories, and so for arbitrary process theories and operational probabilistic theories. 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 re-characterization of quantum supermaps is proven to hold for supermaps on arbitrary convex subsets of channels, including as a special case the supermaps on non-signaling channels used in the study of quantum causal structure.

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Quantum Physics Mathematical Physics Category Theory Mathematical Physics