Proposition: Conditioning requires a probability margin

Let ideal and implemented complete recorded states have trace distance at most $e$. For a retained event let its subnormalized blocks be $X,Y$, of probabilities $p,q$. Then

If $p>e$, then $q>0$ and

No uniform normalized-branch bound follows when $p$ can approach zero.

Proof. Group the record into event and complement. Contraction gives

The second norm is at least $|p-q|$, proving both $|p-q|\le e$ and ${\Vert X-Y\Vert }_1+|p-q|\le 2e$. When $q>0$,

Divide by two and cap at one.