We extend quantum Stein’s lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states \(\rho ^{\otimes n}\) against convex combinations of quantum states \(\sigma ^{\otimes n}\) can be written as a regularized quantum relative entropy formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes an operational interpretation of the relative entropy of coherence in terms of hypothesis testing. For our proof, we start from the composite Stein’s lemma for classical probability distributions and lift the result to the non-commutative setting by using elementary properties of quantum entropy. Finally, our findings also imply an improved recoverability lower bound on the conditional quantum mutual information in terms of the regularized quantum relative entropy—featuring an explicit and universal recovery map.

我们将非对称量子假设检验中的量子 Stein 引理扩展到复合零假设和替代假设。作为我们的主要结果，我们证明了用于测试量子态凸组合\(\rho ^{\otimes n}\)对量子态凸组合\(\sigma ^{\otimes n}\)的渐近误差指数可以写成正则化的量子相对熵公式。我们证明通常需要这样的正则化，但也讨论了我们的公式及其扩展成为单字母的各种设置。这包括在假设检验方面对相干性的相对熵的操作解释。对于我们的证明，我们从经典概率分布的复合 Stein 引理开始，并通过使用量子熵的基本属性将结果提升到非交换设置。最后，我们的研究结果还暗示，在正则化量子相对熵方面，条件量子互信息的可恢复性下界有所提高——具有明确和通用的恢复图。