QEC matrix as an error-vector Gram operator

Let $H_L,H_P,H_O$ be nonzero finite-dimensional complex coordinate spaces with their specified orthonormal bases, and let $V:H_L\to H_P$ be an isometry. Write $\mathrm{\mid }\bar{\mu }⟩=V\mathrm{\mid }\mu ⟩$ for the images of the logical basis vectors. Let $E_l:H_P\to H_O$ be a finite nonempty family of linear maps, and let $H_E$ be a coordinate space with orthonormal basis $\mathrm{\mid }l⟩$ indexed by that family. Define the linear map $A:H_L\otimes H_E\to H_O$ by $A\mathrm{\mid }\mu ,l⟩=E_l\mathrm{\mid }\bar{\mu }⟩$, where $\mathrm{\mid }\mu ,l⟩=\mathrm{\mid }\mu ⟩\otimes \mathrm{\mid }l⟩$. Then $M=A^{†}A$ has entries $M_{[\mu ,l],[\nu ,k]}=⟨\bar{\mu }\mathrm{\mid }{E}_l^{†}{E}_k\mathrm{\mid }\bar{\nu }⟩$ and is positive semidefinite, meaning $x^{†}Mx\ge 0$ for every $x\in H_L\otimes H_E$.

For $v_{\mu ,l}=E_l\mathrm{\mid }\bar{\mu }⟩$, the library's linear-first Gram matrix satisfies $G_{[\mu ,l],[\nu ,k]}=M_{[\nu ,k],[\mu ,l]}$. The partial trace over the first, logical factor satisfies $({\mathrm{Tr}}_{L}M{)}_{l,k}={\sum }_{\mu }{M}_{[\mu ,l],[\mu ,k]}$, retaining the error indices.