In this paper, we extend diagrammatic reasoning in monoidal categories with algebraic operations and equations. We achieve this by considering monoidal categories that are enriched in the category of Eilenberg-Moore algebras for a monad. Under the condition that this monad is monoidal and affine, we construct an adjunction between symmetric monoidal categories and symmetric monoidal categories enriched over algebras for the monad. This allows us to devise an extension, and its semantics, of the ZX-calculus with probabilistic choices by freely enriching over convex algebras, which are the algebras of the finite distribution monad. We show how this construction can be used for diagrammatic reasoning of noise in quantum systems.

翻译：在本文中，我们通过代数运算和方程扩展了幺半范畴中的图表推理。为此，我们考虑了在某个单子的艾伦伯格-摩尔代数范畴中丰富的幺半范畴。在该单子是幺半且仿射的条件下，我们在对称幺半范畴与在该单子代数上丰富的对称幺半范畴之间构造了一个伴随。这使我们能够通过自由地在凸代数（即有限分布单子的代数）上丰富，设计出带概率选择的ZX-演算的扩展及其语义。我们展示了该构造如何用于量子系统中噪声的图表推理。