Categorical supermaps on any generalised theory with channel-state duality are exactly CJ-supermaps, recovering classical, quantum, and NSWSE-Boxworld supermaps.
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Higher order quantum map types are identified with Boolean type functions, with comb types corresponding to chain posets, and type functions decomposed via max/min of basic chains corresponding to affine mixtures and intersections.
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Supermaps on generalised theories
Categorical supermaps on any generalised theory with channel-state duality are exactly CJ-supermaps, recovering classical, quantum, and NSWSE-Boxworld supermaps.
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On the structure of higher order quantum maps
Higher order quantum map types are identified with Boolean type functions, with comb types corresponding to chain posets, and type functions decomposed via max/min of basic chains corresponding to affine mixtures and intersections.