Proposition: Exact flagged transfer normalization

Let $A:C\to R$ and $L:C\to B$ be isometries into occupied subspaces orthogonal to their respective vacuum flags. For $0\le s\le 1$, define

Then

Proof. The accessible channel is

Decode the occupied block exactly and prepare $I_C/d$ on the vacuum block. The decoded channel is $(1-s)\mathrm{id}+s{\mathcal{T}}_{{\tau }_C}$, with error $s{a}_d$.

For any decoder, the vacuum branch of a Bell input has Bell overlap $d^{-2}$; the occupied branch has overlap at most one. The Bell-projector test gives the matching lower bound.

The complementary channel is

Choosing

gives leakage at most $s{a}_d$.

For any other replacement state, project onto the occupied code block and vacuum, sending unused output to vacuum. Let $a$ be its occupied weight. For the Bell input, the effect equal to the occupied Bell projector and zero on vacuum has expectation difference $s-a/d^2$. The effect equal to that projector on the occupied block and identity on vacuum has difference $a(1-d^{-2})$. The first is at least $s{a}_d$ if $a\le s$; the second is at least $s{a}_d$ if $a\ge s$. Thus every replacement channel has distance at least $s{a}_d$.