Polycategorical Constructions for Unitary Supermaps of Arbitrary Dimension

TL;DR

This paper introduces the polyslot construction for symmetric monoidal categories, enabling the composition of quantum supermaps, including infinite-dimensional cases, while preventing time-loops, and applies it to quantum switch examples.

Contribution

It develops a new categorical framework for quantum supermaps that generalizes to infinite dimensions and characterizes key compositional features.

Findings

Polyslot construction enables composition of supermaps in sequence and parallel.

The framework reconstructs quantum supermaps from monoidal structures.

Includes canonical examples like the quantum switch.

Abstract

We provide a construction for holes into which morphisms of abstract symmetric monoidal categories can be inserted, termed the polyslot construction pslot[C], and identify a sub-class srep[C] of polyslots that are single-party representable. These constructions strengthen a previously introduced notion of locally-applicable transformation used to characterize quantum supermaps in a way that is sufficient to re-construct unitary supermaps directly from the monoidal structure of the category of unitaries. Both constructions furthermore freely reconstruct the enriched polycategorical semantics for quantum supermaps which allows to compose supermaps in sequence and in parallel whilst forbidding the creation of time-loops. By freely constructing key compositional features of supermaps, and characterizing supermaps in the finite-dimensional case, polyslots are proposed as a suitable generalization of unitary-supermaps to infinite dimensions and are shown to include canonical examples such as the quantum switch. Beyond specific applications to quantum-relevant categories, a general class of categorical structures termed path-contraction groupoids are defined on which the srep[C] and pslot[C] constructions are shown to coincide.