In this thesis, the author explains the background of problems in quantum estimation, the necessary conditions required for estimation precision benchmarks that are applicable and meaningful for evaluating data in quantum information experiments, and provides examples of such benchmarks. The author develops mathematical methods in quantum estimation theory and analyzes the benchmarks in tests of Bell-type correlation and quantum tomography with those methods. Above all, a set of explicit formulae for evaluating the estimation precision in quantum tomography with finite data sets is derived, in contrast to the standard quantum estimation theory, which can deal only with infinite samples. This is the first result directly applicable to the evaluation of estimation errors in quantum tomography experiments, allowing experimentalists to guarantee estimation precision and verify quantitatively that their preparation is reliable.

This monograph introduces mathematicians, physicists, and engineers to the ideas relating quantum mechanics and symmetries - both described in terms of Lie algebras and Lie groups. The exposition of quantum mechanics from this point of view reveals that classical mechanics and quantum mechanics are very much alike. Written by a mathematician and a physicist, this book is (like a math book) about precise concepts and exact results in classical mechanics and quantum mechanics, but motivated and discussed (like a physics book) in terms of their physical meaning. The reader can focus on the simplicity and beauty of theoretical physics, without getting lost in a jungle of techniques for estimating or calculating quantities of interest.

The purpose of this volume is examine bio-informatics and quantum information, which are growing rapidly at present, and to attempt to connect the two, with a view to enumerating and solving the many fundamental problems they entail. To this end, we look for interdisciplinary bridges in mathematics, physics, and information and life sciences. In particular, research into a new paradigm for information science and life science on the basis of quantum theory is emphasized.

Quantum computation is of great current interest in computer science, mathematics, physical sciences and engineering. The thesis is devoted to the statistical analysis of the model of quantum annealing, and statistical methodologies to construct density matrix estimators. The D-Wave One machine was announced in 2011 as "the world's first commercially available quantum computer" and claimed to run quantum annealing to solve optimization problem. Since the announcement, the model of the D-Wave One machine has been heavily debated. We conduct statistical analysis to compare the result of D-Wave One with the result of the simulated annealing, the simulated quantum annealing and a mean field approximation to quantum annealing. Our statistical analysis shows none of the simulated models fit the D-Wave model well. Meanwhile, comparison of plots and test statistics suggests the model of the D-Wave one is more or less similar to the simulated quantum annealing and the mean field approximation, while different from the simulated annealing. Density matrices describe the quantum states of quantum systems. It is important while difficult to estimate the density matrices, for the elements of density matrices cannot be measured directly. We propose statistical methodologies to construct density matrix estimators from quantum homodyne tomography measurements. We establish an asymptotic theory showing that the proposed density matrix estimators are consistent and have good convergence rates. A numerical study is conducted to demonstrate the finite sample performances of the proposed estimators.

Theory and Implementation. This volume contains papers presented at the NATO Advanced Research Workshop September 9-12, 2009, Quantum Cryptography and Computing Theory and Implementation, that was held in Sopot, Poland and organized by the National Quantum Information Centre of

This book overviews latest ideas and developments in financial econometrics, with an emphasis on how to best use prior knowledge (e.g., Bayesian way) and how to best use successful data processing techniques from other application areas (e.g., from quantum physics). The book also covers applications to economy-related phenomena ranging from traditionally analyzed phenomena such as manufacturing, food industry, and taxes, to newer-to-analyze phenomena such as cryptocurrencies, influencer marketing, COVID-19 pandemic, financial fraud detection, corruption, and shadow economy. This book will inspire practitioners to learn how to apply state-of-the-art Bayesian, quantum, and related techniques to economic and financial problems and inspire researchers to further improve the existing techniques and come up with new techniques for studying economic and financial phenomena. The book will also be of interest to students interested in latest ideas and results.

Quantum-state estimation is an important field in quantum information theory that deals with the characterization of states of affairs for quantum sources. This book begins with background formalism in estimation theory to establish the necessary prerequisites. This basic understanding allows us to explore popular likelihood- and entropy-related estimation schemes that are suitable for an introductory survey on the subject. Discussions on practical aspects of quantum-state estimation ensue, with emphasis on the evaluation of tomographic performances for estimation schemes, experimental realizations of quantum measurements and detection of single-mode multi-photon sources. Finally, the concepts of phase-space distribution functions, which compatibly describe these multi-photon sources, are introduced to bridge the gap between discrete and continuous quantum degrees of freedom.This book is intended to serve as an instructive and self-contained medium for advanced undergraduate and postgraduate students to grasp the basics of quantum-state estimation. Any reader with a solid foundation in quantum mechanics, linear algebra and calculus would be able to follow the book comfortably.

Quantum statistical inference, a research field with deep roots in the foundations of both quantum physics and mathematical statistics, has made remarkable progress since 1990. In particular, its asymptotic theory has been developed during this period. However, there has hitherto been no book covering this remarkable progress after 1990; the famous textbooks by Holevo and Helstrom deal only with research results in the earlier stage (1960s-1970s).This book presents the important and recent results of quantum statistical inference. It focuses on the asymptotic theory, which is one of the central issues of mathematical statistics and had not been investigated in quantum statistical inference until the early 1980s. It contains outstanding papers after Holevo's textbook, some of which are of great importance but are not available now.The reader is expected to have only elementary mathematical knowledge, and therefore much of the content will be accessible to graduate students as well as research workers in related fields. Introductions to quantum statistical inference have been specially written for the book. Asymptotic Theory of Quantum Statistical Inference: Selected Papers will give the reader a new insight into physics and statistical inference.