Quantum-state estimation is an important subject in quantum information theory that focuses on the verification of the quality of a quantum source through statistical inference from data collected using a quantum measurement scheme. The body of literature in this field is still growing, with a substantial amount addressing practical issues that still lack complete solutions. Some novel ideas that have emerged include recent developments in scalable partial estimation schemes for many-body quantum states and quantum channels of large Hilbert-space dimensions, as well as the assignment of error bars for quantum-state estimators.
In order to understand the subject matter well, it is necessary to start from the basics. The book begins with a background formalism for quantum-state estimation to lay the foundation. Next, a substantial amount of time will be spent on exploring some specific estimation schemes, directing the reader’s attention to a few common techniques such as the maximum-likelihood and maximum-entropy estimation schemes, two rather intuitive techniques suitable for an introductory survey on the subject.
After which, the tomographic performance of state estimation schemes as well as other key experimental aspects of quantum-state tomography will be investigated, where the ideas of continuous-variable measurements are introduced. It is also during this time that preparations are made for the final discussion on quasi-probability distribution functions, which are natural representations for measurement data of 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, so that future independent studies can be subsequently carried out with relative ease. Any reader with a solid foundation in quantum mechanics, linear algebra and calculus would be able to follow the book comfortably. The material presented are partly a reflection of my perspectives and experience gained from carrying out research in the field, some of which were presented in 2013 at a summer school hosted in Palack´y University in Olomouc, Czech Republic. In writing this book, I did not follow any other reference material. However, as the book covers aspects of contemporary research, I have supplied a short reading list at the end of every chapter as a source for additional information.
Hints and sample solutions to all problems are included for reference, although they by no means represent the only approaches, only exemplifying ones, since there is typically more than one method to solve a given problem.
A goal of this book is to demonstrate some of the mathematical calculations that are typically performed in the field of quantum-state estimation. Mastering some elementary aspects of complex analysis and special mathematical functions would certainly expedite the learning process for later chapters, although the essential ideas from these topics are reiterated wherever necessary to make the discussions coherent. It is my hope that, after going through the details, the reader will find these mathematical techniques, which are commonly employed in other areas in physics, more easily accessible.