Geometry of Quantum States: An Introduction to Quantum Entanglement

Quantum information theory is at the frontiers of physics, mathematics and information science, offering a variety of solutions that are impossible using classical theory. This book provides an introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. After a gentle introduction to the necessary mathematics the authors describe the geometry of quantum state spaces. Focusing on finite dimensional Hilbert spaces, they discuss the statistical distance measures and entropies used in quantum theory. The final part of the book is devoted to quantum entanglement - a non-intuitive phenomenon discovered by Schrödinger, which has become a key resource for quantum computation. This richly-illustrated book is useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.

Table of Contents:
Preface; 1. Convexity, colours and statistics; 2. Geometry of probability distributions; 3. Much ado about spheres; 4. Complex projective spaces; 5. Outline of quantum mechanics; 6. Coherent states and group actions; 7. The stellar representation; 8. The space of density matrices; 9. Purification of mixed quantum states; 10. Quantum operations; 11. Duality: maps versus states; 12. Density matrices and entropies; 13. Distinguishability measures; 14. Monotone metrics and measures; 15. Quantum entanglement; Epilogue; Appendices; References; Index.

About the Author :
Ingemar Bengtsson is Professor of Physics at Stockholm University. After gaining a Ph.D. in Theoretical Physics from the University of Göteborg (1984), she held post-doctoral positions at CERN, Geneva, and Imperial College, London. She returned to Göteborg in 1988 as a research assistant at Chalmers University of Technnology, before taking up a position as Lecturer in Physics at Stockholm University in 1993. She was appointed Professor of Physics in 2000. Professor Bengtsson is a member of the Swedish Physical Society and a former board member of its Divisions for Particle Physics and for Gravitation. Her favoured research areas are related to geometry, in the forms of general relativity and quantum mechanics. Karol Zyckowski is a Professor at the Institute of Physics, Jagiellonian University, Kraków, Poland and also the Center for Theoretical Physics, Polish Academy of Sciences, Warsaw. He gained his Ph.D. (1987) and habilitation (1994) in theoretical physics at Jagiellonian University, and has followed this with a Humboldt Fellowship in Essen, a Fulbright Fellowship at the University of Maryland, College Park and currently a visiting research position at the Perimeter Institute, Waterloo, Ontario. He has been docent at the Academy of Sciences since 1999 and full professor at Jagiellonian University since 2004. Professor Zyczkowski is a member of the Polish Physical Society and the Institute of Physics. He currently serves on the editorial boards of the journals Open Systems and Information Dynamics and Journal of Physics A.

Review :
'Geometry of Quantum States, not being a quantum mechanics textbook by itself, provides an extensive and detailed insight behind the scenes of entanglement and, as such, can serve as a very useful supplementary text for quantum mechanics courses. Written in a very lucid and engaging style, with numerous illustrations … The spectrum of potential readers … is by no means limited to students and newcomers. It is comprehensive enough to serve as a valuable reference for all researchers interested in quantum information theory. Geometry of Quantum States can be considered an indispensable item on a bookshelf of everyone interest in quantum information theory and its mathematical background.' Milosz Michalski, Open Systems and Information Dynamics 'Bengtsson's and Zyczkowski's book is an artful presentation of the geometry that lies behind quantum theory … the authors collect, and artfully explain, many important results scattered throughout the literature on mathematical physics. The careful explication of statistical distinguishability metrics (Fubini-Study and Bures) is the best I have read.' Gerard Milburn, University of Queensland 'Bengtsson and Zyczkowski's beautifully illustrated volume … attempts to cover considerable ground in its 418 pages.' D. W. Hook, Journal of Physics 'The authors, distinguished mathematical physicists, have written a markedly distinctive, dedicatedly pedagogical, suitably rigorous text, designed, in part, for advanced undergraduates familiar with the principles of quantum mechanics. The book, pleasing in character and enthusiastic in tone, has many stimulating diagrams and tables, as well as problem sets (with hints and answers supplied at the end). The diverse topics covered - conveniently all assembled here - reflect the geometrically-oriented, fundamental quantum-information-theoretic interests and expertise of the two authors.' Paul B. Slater, Mathematical Reviews