Statistical Theory of Quantum Information Processing

从随机数据中推断未知参数，其概率分布依赖于该参数，是统计学中的一个中心问题。这里有几个激发这个提议的问题。我们如何量化存储在原子或光子中的统计信息量，这些原子或光子的状态取决于未知参数？当我们操纵系统时，这些信息是如何改变的？如果我们测量系统并获得由测量结果组成的随机数据，我们可以提取多少信息？该提案的指导原则是量子系统携带固有的统计信息，目的是开发处理这类量子信息的新统计方法。其目标是扩展赫斯特罗姆和霍列沃在七十年代建立的量子统计的框架，在量子工程技术进步和数理统计的现代发展的推动下开辟新的方向。该项目分为三个主要主题。第一个主题旨在发展量子统计决策理论，粗略地说，它是回答激励问题的一般数学框架。这一理论的新奇之处在于，它着眼于量子统计模型的内在结构以及它们之间的所有转换，而不是总是通过测量将它们简化为经典统计模型。第二个主题旨在为量子系统构建更准确的估计工具，这些系统的状态由无限多的参数确定，例如腔中的光。这些技术的灵感来自于非参数统计中最先进的方法，并将通过为实验者提供基于有限数量的测量的准确和可靠的估计值而受益。这与量子工程非常相关，在量子工程中，使用统计估计方法证实了新的和奇异的量子态的创建。第三个主题旨在开发建立连续时间测量的统计方法。这是处理耗散的适当物理框架，耗散是量子计算中主要关注的问题。最近最先进的实验表明，量子反馈控制在量子计量学和量子工程中是一种有价值的工具。然而，人们对统计不确定性如何影响动力学模型的稳定性只有部分了解，参数估计问题的数学分析仍有待开发。总而言之，该项目开辟了横跨数理统计、量子信息和量子滤波与控制的新研究领域，具有深刻而有趣的数学问题，并与实验物理的最新发展密切相关，使其成为一个非常及时的研究计划。

The inference of an unknown parameter from random data whose probability distribution depends on that parameter, is a central topic in Statistics. Here are a few motivating questions of this proposal.How do we quantify the amount of statistical information stored in an atom or a photon whose state depends on an unknown parameter ? How does this information change when we manipulate the system ? How much information can we extract if we measure the system and obtain random data consisting of measurement results ? The guiding principle of the proposal is that quantum systems carry an intrinsic statistical information and the aim is to develop new statistical methods dealing with the processing of this type of quantum information. The goal is to extend the framework of Quantum Statistics established by Helstrom and Holevo in the seventies, by opening new directions motivated by the technological advances in Quantum Engineering and modern developments in Mathematical Statistics. The project is organised in three main topics. The first topic aims at developing a Quantum Statistical Decision Theory which roughly speaking, is the general mathematical framework for answering the motivating questions. The novelty of this theory is that it looks at the intrinsic structure of quantum statistical models and all transformations between them, rather than always reducing them to classical statistical models through measurements. The second topic aims at constructing more accurate estimation tools for quantum systems whose states are determined by an infinite number of parameters, such as the light in a cavity. The techniques are inspired by state-of-the-art methods in non-parametric statistics and will benefit experimentalists by providing them with accurate and reliable estimators based on a limited number of measurements. This is very relevant for Quantum Engineering where the creation of a new and exotic quantum state is confirmed using statistical estimation methods. The third topic aims at developing statistical methods for the set-up of continuous-time measurements. This is the proper physical framework for dealing with dissipation, which is a main concern in Quantum Computation. Recent state-of-the-art experiments have shown that Quantum Feedback Control is a valuable tool in Quantum Metrology and Quantum Engineering. However it is only partially understood how statistical uncertainties influence the stability of the dynamical models, and a mathematical analysis of parameter estimation problems has still to be developed.In conclusion, the project opens a new area of research across Mathematical Statistics, Quantum Information and Quantum Filtering & Control, with deep and interesting mathematical problems and very close connection to the latest developments in Experimental Physics, making it a very timely research programme.