We establish a generalized quantum asymptotic equipartition property (AEP) beyond the i.i.d. framework where the random samples are drawn from two sets of quantum states. In particular, under suitable assumptions on the sets, we prove that all operationally relevant divergences converge to the quantum relative entropy between the sets. More specifically, both the smoothed min- and max-relative entropy approach the regularized relative entropy between the sets. Notably, the asymptotic limit has explicit convergence guarantees and can be efficiently estimated through convex optimization programs, despite the regularization, provided that the sets have efficient descriptions. We give four applications of this result: (i) The generalized AEP directly implies a new generalized quantum Stein's lemma for conducting quantum hypothesis testing between two sets of quantum states. (ii) We introduce a quantum version of adversarial hypothesis testing where the tester plays against an adversary who possesses internal quantum memory and controls the quantum device and show that the optimal error exponent is precisely characterized by a new notion of quantum channel divergence, named the minimum output channel divergence. (iii) We derive a relative entropy accumulation theorem stating that the smoothed min-relative entropy between two sequential processes of quantum channels can be lower bounded by the sum of the regularized minimum output channel divergences. (iv) We apply our generalized AEP to quantum resource theories and provide improved and efficient bounds for entanglement distillation, magic state distillation, and the entanglement cost of quantum states and channels. At a technical level, we establish new additivity and chain rule properties for the measured relative entropy which we expect will have more applications.

翻译：我们在独立同分布框架之外建立了一个广义的量子渐近等分性质（AEP），其中随机样本是从两组量子态中抽取的。特别地，在关于集合的适当假设下，我们证明了所有操作相关的散度都收敛于两组量子态之间的量子相对熵。更具体地说，平滑最小相对熵与最大相对熵均趋近于集合间的正则化相对熵。值得注意的是，尽管存在正则化过程，只要集合具有高效描述，该渐近极限具有显式收敛保证，并可通过凸优化程序进行高效估计。我们给出了该结果的四个应用：（i）广义AEP直接导出了用于在两组量子态之间进行量子假设检验的新广义量子Stein引理。（ii）我们引入了对抗性假设检验的量子版本，其中测试者与拥有内部量子存储器并控制量子设备的对手博弈，并证明最优误差指数可由一种新的量子信道散度概念精确刻画，该概念被命名为最小输出信道散度。（iii）我们推导出相对熵累积定理，表明两个量子信道序列过程之间的平滑最小相对熵，能够以正则化最小输出信道散度之和为下界。（iv）我们将广义AEP应用于量子资源理论，并为纠缠蒸馏、魔术态蒸馏以及量子态与信道的纠缠代价提供了改进的高效边界。在技术层面，我们为测量相对熵建立了新的可加性与链式法则性质，预计这些性质将具有更广泛的应用价值。