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：本文通过代数运算与方程扩展了幺半范畴中的图表推理。我们考虑在伊伦贝格-穆尔代数范畴上富化的幺半范畴（该代数为某单子所定义）。在满足该单子为幺半且仿射的条件下，我们在对称幺半范畴与对称幺半范畴（富化于该单子的代数范畴之上）之间构造了一个伴随关系。通过自由富化于凸代数（即有限分布单子的代数），我们由此构建了带概率选择的ZX-演算扩展及其语义。我们展示了如何利用此构造对量子系统中的噪声进行图表化推理。