Schumacher direct compression achievability

Let $\rho$ be a density operator on a finite-dimensional Hilbert space with von Neumann entropy $H(\rho) = -\operatorname{Tr}(\rho \log \rho)$. For every $\delta > 0$ and $\varepsilon > 0$ there exists $N \in \mathbb{N}$ such that for all block lengths $n \ge N$, the i.i.d. source $\rho^{\otimes n}$ admits a Schumacher compression code of rate at most $H(\rho) + \delta$ with joint (purification) trace-distance error at most $\varepsilon$. Hence $H(\rho)$ is a directly achievable Schumacher compression rate for $\rho$.