This paper studies the difficulty of discriminating quantum channels under operational regimes, proves the quantum channel Stein's lemma (strong converse part), and provides a unified framework to show the operational meaning of quantum channel divergences. First, we establish the exponentially strong converse of quantum channel hypothesis testing under coherent strategies, meaning that any strategy to make the Type II error decays with an exponent larger than the regularized channel relative entropy will unavoidably result in the Type I error converging to one exponentially fast in the asymptotic limit. This result notably delivers the desirable quantum channel Stein's Lemma, enclosing a long-term open problem in quantum information theory. As a byproduct, we show the continuity of the regularized (amortized) Sandwiched R\'{e}nyi channel divergence at $\alpha=1$, resolving another open problem in the field. Second, we develop a framework to show the interplay between the strategies of channel discrimination, the operational regimes, and variants of channel divergences. This framework systematically underlies the operational meaning of quantum channel divergences in quantum channel discrimination. Our work establishes the ultimate limit of quantum channel discrimination, deepening our understanding of quantum channel discrimination and quantum channel divergences in the asymptotic regime. As quantum channel discrimination is strongly connected to many other fundamental tasks in quantum information theory, we expect plentiful applications on related topics such as quantum metrology and quantum communication.
翻译:本文研究在操作制度下歧视量子渠道的困难,证明了斯坦因的量子渠道(强反方部分),并提供了一个统一框架,以显示量子渠道差异的操作意义。首先,我们根据一致的战略建立了量子渠道假设测试的指数强烈反差,这意味着,任何使第二类错误以比正常的渠道相对通则更大的推论而衰减的战略都将不可避免地导致I类误差在无药性限制中迅速趋同于一个指数型的快递。这一结果特别地提供了Stein's Lemma的量子渠道,在量子渠道信息理论中含有一个长期开放的问题。作为副产品,我们展示了量子频道假设(摊合的)R\\{e}nyi类渠道差异在$\alpha=1美元上的延续性反差差,解决该领域的另一个公开问题。第二,我们制定了一个框架,以显示频道歧视战略、操作制度和渠道差异变体之间的相互作用。这个框架系统地体现了我们在量子频道差异的业务意义,在量子渠道歧视中,我们的工作将最终限制质质税歧视的渠道。