Higher-order quantum map types form a distributive lattice of regular subtypes where signalling relations are determined by type function evaluations and structure poset rank parity, with normal forms derived from maximal chains.
On the structure of higher order quantum maps
3 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We study higher order quantum maps in the context of a *-autonomous category of affine subspaces. We show that types of higher order maps can be identified with certain Boolean functions that we call type functions. By an extension of this identification, the algebraic structure of Boolean functions is inherited by some sets of quantum objects including higher order maps. Using the M\"obius transform, we assign to each type function a poset whose elements are labelled by subsets of indices of the involved spaces. We then show that the type function corresponds to a comb type if and only if the poset is a chain. We also devise a procedure for decomposition of the poset to a set of basic chains from which the type function is constructed by taking maxima and minima of concatenations of the basic chains in different orders. On the level of higher order maps, maxima and minima correspond to affine mixtures and intersections, respectively.
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quant-ph 3years
2026 3representative citing papers
Categorical supermaps on any generalised theory with channel-state duality are exactly CJ-supermaps, recovering classical, quantum, and NSWSE-Boxworld supermaps.
Causal reference frame and time-delocalized subsystem descriptions of pure processes are coordinate parametrizations of a single neutral object, with unitary perspective transformations possible by reshuffling time order or adding reference-frame subsystems.
citing papers explorer
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Order structure and signalling in higher order quantum maps
Higher-order quantum map types form a distributive lattice of regular subtypes where signalling relations are determined by type function evaluations and structure poset rank parity, with normal forms derived from maximal chains.
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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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Frame perspectives for process matrices: from coordinate parametrization to spacetime representation
Causal reference frame and time-delocalized subsystem descriptions of pure processes are coordinate parametrizations of a single neutral object, with unitary perspective transformations possible by reshuffling time order or adding reference-frame subsystems.