Statistical Estimation Theory for Quantum Channels

本文基于非对易统计和量子信息几何，研究了未知量子信道的估计问题。主要研究结果如下：1.广义Pauli通道、广义振幅阻尼通道、SU（d）通道等的估计理论：这些通道是量子信息论中常用的标准通道。基于非对易统计、量子信息几何、表示论以及计算机实验数学方法，我们得到了每个量子信道的最优估计方案。我们还阐明了这些估计方案的最优性背后的微分几何机制，特别是输出量子态的量子统计流形关于量子纠缠和扩展程度的膨胀/坍缩.基于局部无偏估计的自适应估计方案的渐近理论：Holevo首先提出的局部无偏估计的概念一直是批评的目标，因为最佳估计方案取决于未知参数本身。为了克服这一困难，Nagaoka提出了一种利用极大似然估计作为试探性估计的自适应估计方案。然而，由于数学上的困难，它的渐近性质的分析一直没有触及。本文利用鞅理论证明了Nagaoka自适应估计方案的强相合性和渐近有效性。

This research addresses the problem of estimating an unknown quantum channel, based on noncommutative statistics and quantum information geometry. Main results are summarized as follows :1. Estimation theory for generalized Pauli channels, generalized amplitude-damping channel, SU(d) channel, etc. :These channels are the standard ones often used in quantum information theory. Based on noncommutative statistics, quantum information geometry, representation theory, as well as methods of experimental mathematics on a computer, we have obtained the optimal estimation scheme for each of those quantum channels. We also have clarified the underlining differential geometrical mechanism, in particular, the dilation/collapse of quantum statistical manifolds of output quantum states with respect to the quantum entanglement and the degree of extensions, behind the optimality of those estimation schemes.2. Asymptotic theory of adaptive estimation schemes based on the locally unbiased estimators :The notion of locally unbiased estimators, advocated first by Holevo, has been the target of criticism because the optimal estimation scheme depends on the unknown parameter itself. In order to surmount this difficulty, Nagaoka advocated an adaptive estimation scheme which make use of maximal likelihood estimators as a tentative estimator. However, due to the mathematical difficulty, the analysis of its asymptotic property has been left untouched. In this research, we have proved the strong consistency and the asymptotic efficiency of Nagaoka's adaptive estimation scheme based on the martingale theory.