On the structure of higher order quantum maps

TL;DR

This paper explores the structure of higher order quantum maps using Boolean functions and poset decompositions, revealing conditions for comb types and methods for their construction.

Contribution

It introduces a novel Boolean function framework for classifying higher order quantum maps and provides a decomposition procedure for their poset structures.

Findings

Type functions correspond to comb types iff the poset is a chain.

Decomposition of posets into basic chains allows construction of type functions.

Maxima and minima of chains relate to affine mixtures and intersections.

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.