QEC matrix and logical partial trace

PDF p. 2: Eq. (6), definitions continuing on p. 3; PDF p. 8: Eq. (S1.6) and following definitions

Let $H_L,H_P,H_O$ be nonzero finite-dimensional complex coordinate spaces with specified orthonormal bases. Let $V:H_L\to H_P$ be a linear isometry, and write $\mathrm{\mid }\bar{\mu }⟩=V\mathrm{\mid }\mu ⟩$ for the code basis, where $\mu =1,\dots ,d_L$ and $d_L=\dim H_L$. Let $E_l:H_P\to H_O$, $l=1,\dots ,r$, be a finite nonempty Kraus family for a quantum channel $\mathcal{N}$, meaning $\mathcal{N}(X)={\sum }_l{E}_{l}X{E}_l^{†}$; a quantum channel is a completely positive, trace-preserving linear map. Let $H_E$ have orthonormal basis $\mathrm{\mid }l⟩$ indexed by the Kraus labels and use the ordered product basis $\mathrm{\mid }\mu ,l⟩=\mathrm{\mid }\mu ⟩\otimes \mathrm{\mid }l⟩$ of $H_L\otimes H_E$.

The QEC matrix $M$ in this basis is defined by $M_{[\mu ,l],[\nu ,k]}=⟨\bar{\mu }\mathrm{\mid }{E}_l^{†}{E}_k\mathrm{\mid }\bar{\nu }⟩$. The adjoint is conjugate transpose in the specified bases, and $⟨u\mathrm{\mid }A\mathrm{\mid }v⟩=u^{†}Av$.

For any operator $B$ on $H_L\otimes H_E$, its logical partial trace ${\mathrm{Tr}}_{L}B$ is the operator on $H_E$ with entries $({\mathrm{Tr}}_{L}B{)}_{l,k}={\sum }_{\mu =1}^{d_L}{B}_{[\mu ,l],[\mu ,k]}$. For a finite square complex matrix $C$, its Frobenius norm is $\mathrm{\parallel }C{\mathrm{\parallel }}_F=({\sum }_{i,j}\mathrm{\mid }{C}_{ij}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$, with the nonnegative square root; equivalently, $\mathrm{\parallel }C{\mathrm{\parallel }}_F^2=\mathrm{Tr}(C^{†}C)$, where the trace is the sum of diagonal entries.

The isometry sends the logical orthonormal basis to an orthonormal code basis. By [F1] and [F3], the given entries are those of the well-defined operator $A^{†}A$ in the specified finite product basis. The construction depends on the chosen code basis and Kraus family, both part of the input.

The logical contraction is the first-factor partial trace in [F4], so it defines an operator on $H_E$. By [F5], the stated nonnegative square root defines the Frobenius norm. For square $C$, matrix multiplication gives $(C^{†}C{)}_{jj}={\sum }_i\overline{C_{ij}}{C}_{ij}$; summing over $j$ proves the displayed trace identity.