The decohered ZX-calculus
Abstract
The discard ZX-calculus is known to be complete and universal for mixed-state quantum mechanics, allowing for both quantum and classical processes. However, if the quantum aspects of ZX-calculus have been explored in depth, little work has been done on the classical side. In this paper, we investigate a fragment of discard ZX-calculus obtained by decohering the usual generators of ZX-calculus. We show that this calculus is universal and complete for affinely supported probability distributions over . To do so, we exhibit a normal form, mixing ideas from the graphical linear algebra program and diagrammatic Fourier transforms. Our results both clarify how to handle hybrid classical-quantum processes in the discard ZX-calculus and pave the way to the picturing of more general random variables and probabilistic processes.
Cite
@article{arxiv.2508.04296,
title = {The decohered ZX-calculus},
author = {Titouan Carette and Daniela Cojocaru and Renaud Vilmart},
journal= {arXiv preprint arXiv:2508.04296},
year = {2025}
}