Fully quantum asymptotic equipartition property

Let $\rho_{AB}$ be a bipartite state on finite-dimensional systems $A$ and $B$, and let $\varepsilon>0$ be the smoothing parameter. For each $n\in\mathbb{N}$, write $\rho_{AB}^{\otimes n}$ for the i.i.d. state on $A^nB^n$. Then, for the entropy quantities $H_{\min}^{\varepsilon}$, $H_{\max}^{\varepsilon}$, and $H$,\[\lim_{\varepsilon\to0}\lim_{n\to\infty}\frac1n H_{\min}^{\varepsilon}(A^n|B^n)_{\rho^{\otimes n}}=H(A|B)_\rho,\qquad\lim_{\varepsilon\to0}\lim_{n\to\infty}\frac1n H_{\max}^{\varepsilon}(A^n|B^n)_{\rho^{\otimes n}}=H(A|B)_\rho.\]