Theorem: Conditional endpoint recovery

Under Definition def:endpoint, let ${\mathcal{N}}_t={\mathrm{Tr}}_{B_{\ast }}{\mathrm{Ad}}_{V_t}$ and ${\mathcal{N}}_{\ast }={\mathrm{Tr}}_{B_{\ast }}{\mathrm{Ad}}_{V_{\ast }}$. Then

Under the external input (eq:ksw),

If $\zeta =0$, there is a single exact endpoint decoder ${\mathcal{D}}_{\ast }$ satisfying

Proof. For common-output isometries $V,W$ and a referenced input state, write the output difference as

with reference identities understood. Its trace norm is at most $2{\Vert V-W\Vert }_{\infty }$. Partial trace contracts trace distance, proving (eq:endpoint-continuity).

For every fixed decoder, postprocessing contraction changes its recovery error by at most ${\nu }_t$. Taking the decoder infimum and using (eq:ksw) at the endpoint proves (eq:endpoint-recovery). If $\zeta =0$, Theorem thm:exact supplies ${\mathcal{D}}_{\ast }$ and contraction gives (eq:fixed-decoder).