(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.

翻译：（对称）幺半群理论通过生成元与等式公理刻画了（对称）幺半群范畴的结构。幺半群理论的项通常采用弦图进行可视化表示。本文引入并研究了一种幺半群理论的定量化版本，其中不再局限于等式推理，而是能够在更抽象的层面上探讨弦图之间的距离关系。这一框架与Mardare等人提出的定量代数理论相呼应，但将其从笛卡尔范畴背景推广至幺半群范畴背景。我们的研究为资源敏感过程（例如量子理论、机器学习、密码学和数字电路理论中的计算过程）的弦图演算进行定量分析奠定了理论基础。