The decohered ZX-calculus

• URL: http://arxiv.org/abs/2508.04296v1
• Date: Wed, 06 Aug 2025 10:32:04 GMT
• Title: The decohered ZX-calculus
• Authors: Titouan Carette, Daniela Cojocaru, Renaud Vilmart,
• Abstract summary: We investigate a fragment of discard ZX-calculus obtained by decohering the usual generators of ZX-calculus.<n>We show that this calculus is universal and complete for affinely supported probability distributions over $mathbbF_2n$.<n>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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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