Lemma: Complete error certificates for the readout components

The quantities in (eq:readcomponents) bound the respective half-diamond distances from ideal damping-free, basis-correct, spectator-identity, and exact-pointer-copy operations.

Proof. For one damping channel, the no-jump operator $L_0$ is a contraction with $\Vert L_0-I\Vert =1-\sqrt{1-r}$. On any input operator, including a reference,

The jump map has complete induced trace norm $r$, since $\Vert L_1{\Vert }^2=r$. Divide the sum by two. Tensor-product telescoping proves the ${\delta }_D$ bound.

For $V_B$, $\Vert V_B-I\Vert \le \frac{1}{2}{\sum }_j|{\alpha }_j|$ by integrating its bounded generator. Lemma lem:Duhamel gives the two remaining bounds.