Transpose fidelity from the QEC matrix

PDF p. 2: Eq. (6); PDF p. 8: Eq. (S1.6)

Let $H_L={\mathbb{C}}^{d_L}$, where $d_L\ge 1$, and let $H_P,H_O$ be nonzero finite-dimensional complex coordinate spaces. Let $V:H_L\to H_P$ be an isometry, $P=V{V}^{†}$, and $\mathcal{N}:\mathcal{L}(H_P)\to \mathcal{L}(H_O)$ a quantum channel with finite nonempty Kraus family $(E_l)$, so $\mathcal{N}(X)={\sum }_l{E}_{l}X{E}_l^{†}$. A quantum channel is a completely positive trace-preserving linear map. Write $\mathrm{\mid }\bar{\mu }⟩=V\mathrm{\mid }\mu ⟩$ for the encoded logical basis and define the QEC matrix by $M_{[\mu ,l],[\nu ,k]}=⟨\bar{\mu }\mathrm{\mid }{E}_l^{†}{E}_k\mathrm{\mid }\bar{\nu }⟩$ in the ordered logical-error product basis.

Put $Q=\mathcal{N}(P)$. Let $Q^{-1\mathrm{/}2}$ be the inverse of its positive square root on its support and zero on its kernel. Let $\mathcal{T}$ be any transpose recovery obtained from the Kraus operators $P{E}_l^{†}{Q}^{-1\mathrm{/}2}$ by completing the map outside the support of $Q$ with a code state. Encode by $V$, apply $\mathcal{N}$ and $\mathcal{T}$, and decode the code-supported output by $V^{†}$; call the resulting logical channel $\mathcal{E}$. Its entanglement fidelity $F_{\mathrm{t}\mathrm{r}}=F_e(\mathcal{E})$ at the maximally mixed input $I\mathrm{/}{d}_L$ is

$F_{\mathrm{t}\mathrm{r}}=d_L^{-2}\mathrm{\parallel }{\mathrm{Tr}}_L\sqrt{M}{\mathrm{\parallel }}_F^2\mathrm{.}$

Here $\sqrt{M}$ is the positive square root, ${\mathrm{Tr}}_L$ contracts the logical indices, and $\mathrm{\parallel }B{\mathrm{\parallel }}_F^2={\sum }_{l,k}\mathrm{\mid }{B}_{l,k}{\mathrm{\mid }}^2$. The formula includes singular $M$ and $Q$ and is independent of the completion choice. The article attributes this expression to Ref. [40].

QEC matrix and logical partial trace

Transpose (Petz) recovery on a code

Entanglement fidelity on a finite logical space