Proposition: Product embeddings preserve the access problem

Let $a:R\to R^{\prime }$ and $b:B\to B^{\prime }$ be isometries, and define

Then

This statement does not authorize a joint relabeling that moves an inaccessible logical register into the accessible factor.

Proof. An isometry $a$ has a CPTP left inverse

where ${\tau }_R$ is any fixed state on $R$. A decoder on $R^{\prime }$ restricts to one on $R$ by composition with ${\mathrm{Ad}}_a$; a decoder on $R$ extends to $R^{\prime }$ by composition with $L_a$. These two operations give the recovery equality.

For leakage, embedding a replacement state by $b$ gives one inequality. Applying $L_b$ to an arbitrary candidate replacement state on $B^{\prime }$ gives the other inequality by contraction. A joint transformation mixing $R$ and $B$ is not of the stated product form and need not preserve either optimization problem.