Brandão's generalised quantum Stein's lemma

Let $(\mathcal{M}_n)_n$ be a family of sets of quantum states satisfying Axioms 1--5 in Section~, and let $D^\infty_{\mathcal{M}}(\rho)$ denote the corresponding regularised relative entropy. Brandão's generalised quantum Stein's lemma. For every quantum state $\rho$,

$$\lim_{\epsilon \to 0} \lim_{n \to \infty} \frac{1}{n} \min_{\sigma_n \in \mathcal{M}_n} D^\epsilon_H\left(\rho^{\otimes n} \middle\| \sigma_n\right)=D^\infty_{\mathcal{M}}(\rho).$$

This extends quantum Stein's lemma from a single i.i.d. alternative state to composite hypothesis testing against families of states, and connects the asymptotic hypothesis-testing exponent with the regularised relative entropy. The supplied text presents it as the seminal result of Brandão and does not provide evidence resolving the candidate's status.