Theorem: Exact arbitrary-code recovery and residual leakage

For the complete pre-endpoint prefix,

The maximum Bell-input entanglement fidelity is

An explicit optimal decoder reads $r$, locates the occupied transfer block without reading its logical basis, applies ${U_r}^{†}$ to that block, and returns it. On the all-vacuum pending branch it prepares $I_d/d$.

Proof. The stated decoder has complete decoded channel

Lemma lem:depolarization gives error $s_t{c}_d$.

For any decoder, a completed branch contributes Bell overlap at most one. The accessible pending branch is input-independent. Its reference state remains $I_S/d$, so after any local decoding its Bell overlap is $d^{-2}$. Therefore

The Bell-projector lower bound proves that no decoder has error smaller than $s_t{c}_d$, and the stated decoder attains both optima.

For leakage, choose the constant state $\tau ={\mathcal{C}}_t(I_d/d)$. The completed summands cancel in ${\mathcal{C}}_t-{\mathcal{T}}_{\tau }$. On the pending summands the difference is $s_t$ times a reversible flagged Weyl encoding of ${\mathrm{id}}_d-{\mathcal{T}}_d$. Contractivity, followed by its left inverse, gives the upper bound $s_t{c}_d$.

To prove optimality against an arbitrary constant state, apply a CPTP map to the residual that retains a pending/completed flag, corrects $U_r$ and returns the logical qudit on pending branches, and returns a fixed state on completed branches. On the Bell input, the pending block of the actual state is $s_t|{\Phi }_d⟩⟨{\Phi }_d|$. Let $a$ be the pending weight of the mapped candidate constant state. Its pending Bell expectation is $a/d^2$.

Use first the effect equal to the Bell projector on pending and zero on completed. Its expectation difference is $s_t-a/d^2$. Use next the effect equal to that Bell projector on pending and identity on completed. Its expectation difference is $a(1-d^{-2})$. If $a\le s_t$, the first is at least $s_t{c}_d$; if $a\ge s_t$, the second is at least $s_t{c}_d$. The trace-distance effect bound, and hence the diamond norm, gives the required lower bound for every constant state.