Quantitative Monoidal Algebra: Axiomatising Distance with String Diagrams

(Symmetric) monoidal theories encapsulate presentations by generators and equations for (symmetric) monoidal categories. Terms of a monoidal theory are typically represented pictorially using string diagrams. In this work we introduce and study a quantitative version of monoidal theories, where instead of equality one may reason more abstractly about distance between string diagrams. This is in analogy with quantitative algebraic theories by Mardare et al., but developed in a monoidal rather than cartesian setting. Our framework paves the way for a quantitative analysis of string diagrammatic calculi for resource-sensitive processes, as found e.g. in quantum theory, machine learning, cryptography, and digital circuit theory.