Factor-two guarantee for transpose recovery

PDF p. 2: Checking-time paragraph preceding Eq. (6)

Let $d\ge 1$ be an integer, let ${\mathbb{C}}^d$ have its standard inner product, and let $H_P,H_O$ be nonzero finite-dimensional complex Hilbert spaces. Let $V:{\mathbb{C}}^d\to H_P$ be a linear isometry onto the code, with code projector $P=V{V}^{†}$. Let $\mathcal{N}:\mathcal{L}(H_P)\to \mathcal{L}(H_O)$ be a finite-dimensional quantum channel, meaning a completely positive trace-preserving linear map. Write ${\mathrm{Ad}}_V(X)=VX{V}^{†}$.

For a logical channel $\mathcal{E}$, let $F_e(\mathcal{E})$ be its entanglement fidelity at the maximally mixed input $I\mathrm{/}d$: the overlap of a normalized maximally entangled logical-reference state with its image under $\mathcal{E}$ on the logical factor and the identity on the reference. Let $F_{\mathrm{o}\mathrm{p}\mathrm{t}}={\max }_{\mathcal{R}}{F}_e(\mathcal{R}\mathcal{N}{\mathrm{Ad}}_V)$, where $\mathcal{R}$ ranges over all completely positive trace-preserving logical decoders from $\mathcal{L}(H_O)$ to $\mathcal{L}({\mathbb{C}}^d)$.

Let $F_{\mathrm{t}\mathrm{r}}$ be the same fidelity for a support-completed transpose recovery followed by compression to the logical space. Explicitly, for Kraus operators $E_a$ of $\mathcal{N}$, the transpose map has Kraus operators $P{E}_a^{†}\mathcal{N}(P{)}^{-1\mathrm{/}2}$, with the inverse square root taken on the support and zero on the kernel; it is completed outside that support to a code-supported channel. Then

$1-F_{\mathrm{o}\mathrm{p}\mathrm{t}}\le 1-F_{\mathrm{t}\mathrm{r}}\le 2(1-F_{\mathrm{o}\mathrm{p}\mathrm{t}})\mathrm{.}$

The bound concerns the fixed maximally mixed input and does not assert a worst-case pure-state fidelity bound.

The article invokes this guarantee to motivate checking time as reflecting the code rather than a particular decoder. This contextual interpretation is attributed to the article; the mathematical deduction above uses the independent near-optimality supplier.