On Traces in Categories of Contractions

TL;DR

This paper generalizes the concept of traced monoidal categories, showing that categories with Moore-Penrose pseudoinverses support feedback loops, extending previous results from Hilbert spaces to a broader categorical context.

Contribution

It demonstrates that Bartha's result on traced subcategories of isometries and contractions extends to any dagger additive category with Moore-Penrose pseudoinverses.

Findings

Traced monoidal structure applies to categories with Moore-Penrose pseudoinverses.

The result generalizes previous work from Hilbert spaces to broader categories.

Supports modeling of feedback in quantum and other processes.

Abstract

Traced monoidal categories are used to model processes that can feed their outputs back to their own inputs, abstracting iteration. The category of finite dimensional Hilbert spaces with the direct sum tensor is not traced. But surprisingly, in 2014, Bartha showed that the monoidal subcategory of isometries is traced. The same holds for coisometries, unitary maps, and contractions. This suggests the possibility of feeding outputs of quantum processes back to their own inputs, analogous to iteration. In this paper, we show that Bartha's result is not specifically tied to Hilbert spaces, but works in any dagger additive category with Moore-Penrose pseudoinverses (a natural dagger-categorical generalization of inverses).