A Yoneda lemma for categorical supermaps gives a concrete representation via channel-state duality whenever the theory has it, yielding stable definitions for boxworld and real quantum theory.
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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
A Yoneda lemma for categorical supermaps gives a concrete representation via channel-state duality whenever the theory has it, yielding stable definitions for boxworld and real quantum theory.
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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.