Lemma: Local embeddings do not change unrestricted recovery or complete leakage

With the access-preserving embeddings of Definition def:access,

where

Proof. Any isometry $j$ has a CPTP left inverse

for a fixed state $\tau$ on its domain. A decoder after ${\mathrm{Ad}}_j\mathcal{N}$ composes with ${\mathrm{Ad}}_j$ to give a decoder after $\mathcal{N}$; a decoder after $\mathcal{N}$ composes with $L_j$ to give a decoder after ${\mathrm{Ad}}_j\mathcal{N}$. The resulting decoded channels coincide, so their optimal errors coincide.

For the complement, embedding any replacement state gives

because the output isometry preserves trace norms, including references. This proves one direction. For any replacement state $\omega$ on the larger space, postprocessing by $L_k$ gives

Taking infima proves the opposite direction.