Lemma: Operator approximation to encoded channel approximation

Suppose finite- and infinite-volume dynamics obey

for every operator $O$ in the declared finite detector algebra. Suppose their initial preparations agree on the local restrictions used in this comparison. For a code of dimension $d$,

The estimate includes arbitrary code references.

Proof. A reference of dimension $d$ suffices. For a detector–reference observable $Z$ with $\Vert Z\Vert \le 1$, write its reference blocks as $Z_{ij}$. Each block has norm at most one.

The evolved difference is a $d\times d$ block matrix, with every block bounded by ${\eta }_L(t)$. For a block vector $v=(v_j)$,

Cauchy–Schwarz, followed by summation over $i$, gives $\Vert \Delta Z\Vert \le d{\eta }_L(t)$.

Evaluate this operator in the common prepared reference–source–bath state. Trace-norm duality gives an output trace-norm difference at most $d{\eta }_L(t)$. Taking the input supremum and dividing by two proves the claim.