Quantum channel estimation and discrimination are fundamentally related information processing tasks of interest in quantum information science. In this paper, we analyze these tasks by employing the right logarithmic derivative Fisher information and the geometric Rényi relative entropy, respectively, and we also identify connections between these distinguishability measures. A key result of our paper is that a chain-rule property holds for the right logarithmic derivative Fisher information and the geometric Rényi relative entropy for the interval \(\alpha \in (0,1) \) of the Rényi parameter \(\alpha \). In channel estimation, these results imply a condition for the unattainability of Heisenberg scaling, while in channel discrimination, they lead to improved bounds on error rates in the Chernoff and Hoeffding error exponent settings. More generally, we introduce the amortized quantum Fisher information as a conceptual framework for analyzing general sequential protocols that estimate a parameter encoded in a quantum channel. We then use this framework, beyond the aforementioned application, to show that Heisenberg scaling is not possible when a parameter is encoded in a classical–quantum channel. We then identify a number of other conceptual and technical connections between the tasks of estimation and discrimination and the distinguishability measures involved in analyzing each. As part of this work, we present a detailed overview of the geometric Rényi relative entropy of quantum states and channels, as well as its properties, which may be of independent interest.

量子信道估计和判别是量子信息科学中从根本上相关的信息处理任务。在本文中，我们分别通过使用正确的对数导数Fisher信息和几何Rényi相对熵来分析这些任务，并且还确定了这些可区分性度量之间的联系。本文的关键结果是，对于右对数导数Fisher（）的右对数导数Fisher信息和对数Rényi参数\（\ （\ alpha \ in（0,1）\）的几何Rényi相对熵，其链规则性质成立。 α \）。在信道估计中，这些结果暗示了海森堡缩放无法达到的条件，而在信道识别中，它们导致了Chernoff和Hoeffding错误指数设置中错误率的改善边界。更一般而言，我们将摊销的量子Fisher信息引入作为概念框架，用于分析估计在量子通道中编码的参数的一般顺序协议。然后，我们使用上述框架（超出了前述应用程序）来证明，当参数在经典量子通道中编码时，海森堡缩放是不可能的。然后，我们确定了估计和歧视任务与分析每个任务所涉及的区别性度量之间的许多其他概念和技术联系。作为这项工作的一部分，