Theorem: Whole ambient code: exact errors

For every fixed $U$, at a cut with $r=r_k$, $b=b_k$, the coherent channels satisfy

If the returned radiation is destructively read out, its classical channel satisfies

In particular, a midpoint mean-entropy peak does not imply accurate recovery of the whole $D$-dimensional input.

Proof. Conjugating the logical target by $U$ leaves the optimization unchanged. The coherent channel becomes ${\mathrm{Tr}}_B$. The decoder appending $I_B/b$ has error $1-b^{-2}$, by the Weyl twirl on $B$. For the reverse bound, use the product Bell input ${\Phi }_r\otimes {\Phi }_b$. After $B$ is lost, its reference is independent of every decoder input. The decoded full Bell overlap is at most $b^{-2}$. This proves the recovery formula.

The complementary channel is ${\mathrm{Tr}}_R$. Replacement by $I_B/b$ has distance at most $1-b^{-2}$. Fixing the input on $R$ and testing a Bell input on $B$ gives the matching lower bound for every replacement.

For record-only access, the measurement effects on the transformed code are

If ${\sigma }_y$ is any decoder output, its full Bell overlap is

The decoder preparing $|y⟩⟨y{|}_R\otimes I_B/b$ attains the bound: its channel is ${\Delta }_R\otimes {\mathcal{T}}_{I_B/b}$, the average of $r{b}^2$ phase/Weyl conjugations, one of which is the identity. Separating that term proves the matching upper bound.