Proposition: Conditional repeated-use reduction

Fix $(\gamma ,\nu )$. If (eq:population-bound) holds for every $n$ and every block input, then

Every maximizing phase-pair product ensemble attains the block Holevo value.

Proof. For an arbitrary ensemble of block signals, let ${\overline{p}}_j$ be its mean population at site $j$. Diagonal entropy and classical subadditivity bound the mean output entropy above by

The assumed bound and convexity of $e$ bound the average conditional output entropy below by ${\sum }_{j}e({\overline{p}}_j)$. Subtracting gives at most $n{\max }_p{F}_{\gamma ,\nu }(p)$. Theorem thm:oneuse supplies a product ensemble attaining it. Equation (eq:regularized) then gives the capacity identity.