Adaptiveness is a key principle in information processing including statistics and machine learning. We investigate the usefulness of adaptive methods in the framework of asymptotic binary hypothesis testing, when each hypothesis represents asymptotically many independent instances of a quantum channel, and the tests are based on using the unknown channel and observing outputs. Unlike the familiar setting of quantum states as hypotheses, there is a fundamental distinction between adaptive and non-adaptive strategies with respect to the channel uses, and we introduce a number of further variants of the discrimination tasks by imposing different restrictions on the test strategies. The following results are obtained: (1) We prove that for classical-quantum channels, adaptive and non-adaptive strategies lead to the same error exponents both in the symmetric (Chernoff) and asymmetric (Hoeffding, Stein) settings. (2) The first separation between adaptive and non-adaptive symmetric hypothesis testing exponents for quantum channels, which we derive from a general lower bound on the error probability for non-adaptive strategies; the concrete example we analyze is a pair of entanglement-breaking channels. (3)We prove, in some sense generalizing the previous statement, that for general channels adaptive strategies restricted to classical feed-forward and product state channel inputs are not superior in the asymptotic limit to non-adaptive product state strategies. (4) As an application of our findings, we address the discrimination power of an arbitrary quantum channel and show that adaptive strategies with classical feedback and no quantum memory at the input do not increase the discrimination power of the channel beyond non-adaptive tensor product input strategies.

翻译：自适应是信息处理（包括统计和机器学习）中的关键原则。我们在渐近二元假设检验框架下研究自适应方法的实用性，其中每个假设代表渐近多个独立的量子信道实例，测试基于使用未知信道并观察输出。与熟悉的以量子态作为假设的设定不同，在信道使用方面，自适应策略与非自适应策略之间存在根本性区别，我们通过施加不同的测试策略限制，引入了区分任务的多种变体。获得以下结果：（1）我们证明，对于经典-量子信道，在对称（Chernoff）和非对称（Hoeffding、Stein）设定下，自适应和非自适应策略导致相同的误差指数。（2）我们首次在量子信道的对称假设检验指数上实现自适应与非自适应策略的分离，这基于非自适应策略错误概率的通用下界；我们分析的具体例子是一对纠缠破缺信道。（3）我们在某种意义上推广了上述结论，证明对于一般信道，限制为经典前馈和乘积态信道输入的自适应策略在渐近极限下并不优于非自适应乘积态策略。（4）作为我们发现的应用，我们探讨了任意量子信道的区分能力，并表明带有经典反馈但无输入量子内存的自适应策略，其信道区分能力并未超越非自适应张量积输入策略。