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 R\'enyi 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 R\'enyi relative entropy for the interval $\alpha\in(0,1) $ of the R\'enyi 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, and we 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 R\'enyi relative entropy of quantum states and channels, as well as its properties, which may be of independent interest.
翻译:量子信道估计和区别从根本上说是量子信息科学中感兴趣的信息处理任务。 在本文中,我们通过分别使用正确的对数衍生渔业衍生物信息以及海森堡测量值相对对流信息来分析这些任务,我们还确定了这些可辨别措施之间的联系。我们论文的一个重要结果是,对于正确的对数衍生物渔业信息,以及R'enyi相对对流信息,存在着一种链式属性,这是对数衍生物渔业信息与R'enyi 相对映射值之间的一个概念框架,用于对量子频道中编码的参数进行估计(0.1美元 ) 。在频道估计中,这些结果意味着一个条件使得海森堡测量值无法实现,同时在频道歧视中,这些结果导致改进了切诺夫和霍菲错误推断值设置的误率界限。更一般而言,我们引入了摊分量级量渔业信息,作为概念框架,用以评估量子频道中标定值参数的值值(0.1美元 ) 。我们使用这个框架,以显示海森堡测量的缩度是不可能实现的,而当一个参数的不易测度的,而我们将这个参数的相对连接的参数作为一个概念和直径段段内分析的相对分析,我们所参与的参数作为分析的一个。