Schumacher witness-to-achievability bridge

Let $\rho$ be a density operator on a finite-dimensional Hilbert space with von Neumann entropy $H(\rho) = -\operatorname{Tr}(\rho \log \rho)$. If a typical-compression witness family supplies, for every $\delta, \varepsilon > 0$, an $N \in \mathbb{N}$ such that every blocklength $n \ge N$ has a Schumacher compression code for $\rho^{\otimes n}$ with rate at most $H(\rho)+\delta$ and error at most $\varepsilon$, then $H(\rho)$ is a directly achievable Schumacher compression rate for $\rho$. This bridge theorem assumes the typical-compression witness family; unconditional direct achievability is tracked separately.