Infrared Safety from ZX-Diagrams: A Categorical Proof of Soft-QED as Open Quantum System
2026-06-29 • Logic in Computer Science
Logic in Computer Science
AI summaryⓘ
The authors used a visual language called discard ZX-calculus to prove a complex quantum physics idea called the Bloch-Nordsieck cancellation, which deals with infinite results in quantum electrodynamics (QED). They treated soft photons (very low energy photons) as part of an environment that affects the main particles and higher energy photons, using tools from open quantum systems. Their proof simplifies complicated quantum behavior into a single diagram, confirming earlier physical arguments and showing how soft photons cause decoherence. They also created new methods to verify certain properties in quantum simulations, connecting abstract math with practical quantum computing.
discard ZX-calculusBloch-Nordsieck cancellationquantum electrodynamics (QED)soft photonsopen quantum systemsCPTP mapsoft-photon theoremFeynman-Vernon influence functionalLindblad generatorprocess tensors
Authors
Soo-Jong Rey
Abstract
The discard ZX-calculus, a diagrammatic language for mixed-state quantum mechanics, is used to give a nonperturbative, categorical proof of the Bloch-Nordsieck cancellation of infrared divergences in QED. Soft photons are treated as an open quantum system: the resolved charged particles and hard photons form the system, while photons below a detector resolution form the environment. The reduced hard channel is a completely positive trace-preserving (CPTP) map, and the soft-photon theorem replaces the full S-matrix by a controlled displacement operator whose Feynman-Vernon influence functional satisfies the equal-history normalization ${\cal F}[J,J]=1 $. In the ZX-calculus, this normalization is a single diagrammatic identity: the doubled displacement diagram collapses to the bare wire under the unitarity, cyclicity, and discard rules. The proof therefore serves as a categorical consistency check on the open-system treatment of soft QED given in a companion paper; it confirms that the physical derivation is logically complete and free of hidden assumptions about the infrared limit. For off-diagonal hard-state elements, the same diagram yields the coherent-state overlap, giving a first-principles account of soft-cloud decoherence. The soft-shell coarse graining is then constructed as a CPTP Schur channel whose infinitesimal limit produces the exact Lindblad generator with jump operators determined by the eikonal emission amplitudes. Finally, a local CPTP-certification pipeline is developed for non-Markovian process tensors, enabling constant-time verification of trace preservation in open quantum simulations. The framework bridges categorical quantum semantics, non-equilibrium field theory, and practical open-system compilation.