There is a bijection between odd prime dimensional qudit pure stabilizer states modulo invertible scalars and affine Lagrangian subspaces of finite dimensional symplectic $\mathbb{F}_p$-vector spaces. In the language of the stabilizer formalism, full rank stabilizer tableaus are exactly the bases for affine Lagrangian subspaces. This correspondence extends to an isomorphism of props where the composition of stabilizer circuits becomes the relational composition of affine subspaces and the tensor product becomes the direct sum. In this paper, we extend this correspondence between stabilizer circuits and tableaus to the mixed setting; by regarding stabilizer codes as affine coisotropic susbspaces (again only in odd prime qudit dimension/for qubit CSS codes). We show that by splitting the projector for a stabilizer code we recover the error detection protocol and the error correction protocol with affine classical processing power.

翻译：存在一个双射，将奇素数维qudit纯稳定子态（模可逆标量）与有限维辛$\mathbb{F}_p$-向量空间的仿射拉格朗日子空间一一对应。在稳定子形式体系的语言中，满秩稳定子Tableau恰为仿射拉格朗日子空间的基。这一对应可扩展为props同构，其中稳定子电路的复合变为仿射子空间的关系复合，张量积变为直和。本文将此对应从稳定子电路和Tableau扩展至混合情形：通过将稳定子码视为仿射余迷向子空间（同样仅适用于奇素数维qudit/量子比特CSS码）。我们证明，通过分裂稳定子码的投影算子，可重构含仿射经典处理能力的错误检测协议与错误纠正协议。