Statistical Theory of Quantum Information Processing

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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.