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 $\mathbb{F}_{2}^{n}$. 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.