Lemma: Finite POVM comparison

Let $\{F_x\}$ be a physical POVM, $\{G_x\}$ a logical POVM, and $J$ an isometry. If

for every outcome, then for every logical state $\rho$,

If the prepared physical state is within trace distance $e_{\mathrm{prep}}$ of $J\rho J^{†}$, add $e_{\mathrm{prep}}$ to the allowance.

Proof. Each outcome probability differs by at most $a_x$. Sum their absolute differences and divide by two. Measurement contracts preparation trace distance, so the second assertion follows by the triangle inequality.