Lemma: An erasure isometry and its exact errors

Let $J:C\to A$ and $K:C\to B$ be isometries whose ranges are orthogonal to fixed unit vectors $a_0$ and $b_0$. For $r\in [0,1]$, define

Its accessible channel and complement are

and

An optimal decoder and residual comparison state are

At a zero-weight endpoint, the unused embedding need not be defined.

Proof. Orthogonality gives the two partial traces. The decoder is normal, completely positive, and trace preserving, and

Lemma lem:replacement gives error $r{c}_d$.

For any decoder on a Bell input, the successful branch contributes Bell overlap at most $1-r$. The erased branch contributes exactly $r/d^2$. The Bell test proves recovery optimality.

For the stated residual state,

which gives the leakage upper bound.

For any alternative $\tau$, apply a CPTP map to the residual that keeps the $K$ range as a logical sector and maps its orthogonal complement to a single failure flag. Let $t$ be the candidate state's weight in the logical sector. On the Bell input, the effect equal to the logical Bell projector and zero on the failure sector has expectation difference

The effect equal to that Bell projector plus the entire failure sector has expectation difference

If $t\le r$, the first is at least $r{c}_d$. If $t\ge r$, the second is at least $r{c}_d$. Contractivity proves the lower bound for every $\tau$. Endpoints follow from the constant or isometric channels.