Theorem: C : anchoring and mesh control

Let $\mathsf{S}\in {\mathcal{A}}_N$, let $n=\left|L\right|$, and let $s=\left|S\right|\ge 1$.

1. Every diagonal measured reduction whose weights belong to ${\mathcal{R}}_{\mathrm{cert}}(N,s)$ is exactly Born.

2. If $n+s\le N$, then

   $$d_{\mathrm{TV}}({{\mathsf{P}}^{\mathsf{S}}}_{\psi },{\mathrm{Born}}_{\psi }^S)\le {\omega }_N\left(\sqrt{2{\delta }_{\mathrm{cert}}(N,s)}\right)+{\delta }_{\mathrm{cert}}(N,s).$$

   The same bound holds with the fine-graining net and its radius.

Proof. Part (i) is Propositions prop:fg-anchor and prop:symmetry, including purification locality.

For (ii), use Lemma lem:record-marginal. Write the copied vector with the record first as

where the $v_{\kappa }$ are orthonormal on positive weights. The environment has $n\ge s$ qubits, so extend this family orthonormally to every record label. Choose $q$ in the certified net with $d_{\mathrm{TV}}(p,q)\le {\delta }_{\mathrm{cert}}$, and define

Then

The $R$-law at ${\Psi }_q$ is $q$ by (i). Uniform continuity and the triangle inequality give

The record identity transfers this to direct $S$-readout.