Proposition: Instrument and conditional-state stability

For the fixed flagged decoder,

For every common referenced input, each flag probability differs by at most $\xi$.

Let ${\tau }_y,{\hat{\tau }}_y$ be corresponding unnormalized conditional reference–qubit outputs. If both branch probabilities are at least $p_{\min }>0$, then

No uniform normalized-branch conclusion follows as the branch probability tends to zero.

Proof. The first assertion is CPTP postprocessing contraction. Flag probabilities are effect expectations.

The flagged output is block diagonal, so ${\Vert {\hat{\tau }}_y-{\tau }_y\Vert }_1\le 2\xi$. Writing $p=\mathrm{Tr}{\tau }_y$, $q=\mathrm{Tr}{\hat{\tau }}_y$,

Divide by two, use $q\ge p_{\min }$, and cap state distance at one.