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式推理。我们考虑在函子范畴的艾伦伯格-穆尔代数上丰富的幺半范畴来实现这一目标。在函子满足幺半与仿射条件的前提下，我们在对称幺半范畴与基于该函子代数丰富的对称幺半范畴之间构造了一个伴随对。由此，我们通过自由地在凸代数（有限分布函子的代数）上丰富，设计出包含概率选择的ZX-演算扩展及其语义。我们展示了如何将该构造用于量子系统中噪声的图式推理。