Remark: Explicit entropy check

For an unreferenced message state ${\rho }_M$, let $s=S({\tau }_B)=H(p)$. The final state satisfies

The logical $AB$ state is the pure pair and satisfies $S(B|A)=-s$. Their sum is $-2s$, but this is not a violation of Lemma lem:weak: $A$ and $E$ are not independent factors. There are two descriptions of overlapping observables, not two purifiers on independent systems.