Higher-Order Quantum Objects are Strong Profunctors

TL;DR

This paper establishes a formal connection between higher-order quantum maps and strong profunctors, demonstrating how causality constraints can be expressed within a profunctorial framework over symmetric monoidal categories.

Contribution

It constructs a functor linking higher-order causal categories to strong profunctors, generalizing quantum theory to broader categorical contexts.

Findings

The functor F is full, faithful, and strongly closed when C is additive.

The embedding captures one-way signalling constraints via coend calculus.

Profunctorial approach extends higher-order quantum theory to symmetric monoidal categories.

Abstract

We explore the sense in which the existing constructions for higher-order maps on quantum theory based on causality constraints and compositionality constraints respectively, coincide. More precisely, we construct a functor F : Caus(C) -> StProf(C1) from higher-order causal categories to the category of strong profunctors over first-order causal processes that is lax-lax duoidal, full, faithful, and strongly closed whenever C is additive. When C = CP this embedding is furthermore strong on the sequencer for duoidal categories, expressing the possibility to interpret one-way signalling (but not general non-signalling) constraints in terms of the coend calculus for profunctors. We conclude that insofar as compositional constraints can be used to express causality constraints, the profunctorial approach generalises higher-order quantum theory to a construction over general symmetric monoidal categories.