Quantum hypothesis testing has been traditionally studied from the information-theoretic perspective, wherein one is interested in the optimal decay rate of error probabilities as a function of the number of samples of an unknown state. In this paper, we study the sample complexity of quantum hypothesis testing, wherein the goal is to determine the minimum number of samples needed to reach a desired error probability. By making use of the wealth of knowledge that already exists in the literature on quantum hypothesis testing, we characterize the sample complexity of binary quantum hypothesis testing in the symmetric and asymmetric settings, and we provide bounds on the sample complexity of multiple quantum hypothesis testing. In more detail, we prove that the sample complexity of symmetric binary quantum hypothesis testing depends logarithmically on the inverse error probability and inversely on the negative logarithm of the fidelity. As a counterpart of the quantum Stein's lemma, we also find that the sample complexity of asymmetric binary quantum hypothesis testing depends logarithmically on the inverse type~II error probability and inversely on the quantum relative entropy. Finally, we provide lower and upper bounds on the sample complexity of multiple quantum hypothesis testing, with it remaining an intriguing open question to improve these bounds.

翻译：量子假设检验传统上从信息论角度进行研究，重点在于作为未知状态样本数量函数的错误概率最优衰减率。本文研究量子假设检验的样本复杂度问题，目标在于确定达到目标错误概率所需的最小样本数量。通过利用量子假设检验文献中已有的丰富知识，我们在对称与非对称场景下刻画了二元量子假设检验的样本复杂度，并给出了多元量子假设检验的样本复杂度界限。具体而言，我们证明对称二元量子假设检验的样本复杂度与错误概率倒数的对数呈对数关系，与保真度的负对数呈反比关系。作为量子Stein引理的对应结果，我们还发现非对称二元量子假设检验的样本复杂度与第二类错误概率倒数的对数呈对数关系，与量子相对熵呈反比关系。最后，我们给出了多元量子假设检验样本复杂度的上下界，而如何改进这些界限仍是一个值得探讨的开放性问题。