The generalized quantum Stein's lemma for resource theories

Let $\{\mathcal F_n\}_{n\in\mathbb N}$ be a family of sets of quantum states satisfying Axioms 1–5, and let $\rho$ be a quantum state. For $\varepsilon>0$, write $D_H^\varepsilon(\rho\|\sigma)$ for the hypothesis-testing relative entropy, and define the regularized relative entropy of resource by

$$D_\mathcal F^\infty(\rho)=\lim_{n\to\infty}\frac{1}{n}\inf_{\sigma\in\mathcal F_n}D(\rho^{\otimes n}\|\sigma).$$

Generalized quantum Stein's lemma. For any such family,

$$\lim_{\varepsilon\to0}\liminf_{n\to\infty}\frac{1}{n}\min_{\sigma_n\in\mathcal F_n}D_H^\varepsilon(\rho^{\otimes n}\|\sigma_n)=D_\mathcal F^\infty(\rho).$$

This conjecture extends quantum Stein's lemma from a fixed alternative state to convex sets of quantum states in quantum resource theories. The statement is established for quantum coherence when $\mathcal N$ is the diagonal subalgebra, while the general case under Axioms 1–5 remains open.