Geometry of Quantum States

Quantum information theory is a branch of science at the frontier of physics, mathematics, and information science, and offers a variety of solutions that are impossible using classical theory. This book provides a detailed introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. The second edition contains new sections and entirely new chapters: the hot topic of multipartite entanglement; in-depth discussion of the discrete structures in finite dimensional Hilbert space, including unitary operator bases, mutually unbiased bases, symmetric informationally complete generalized measurements, discrete Wigner function, and unitary designs; the Gleason and Kochen–Specker theorems; the proof of the Lieb conjecture; the measure concentration phenomenon; and the Hastings’ non-additivity theorem. This richly-illustrated book will be 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

Chapter 1: Convexity, Colours And Statistics

Chapter 2: Geometry Of Probability Distributions

Chapter 3: Much Ado About Spheres

Chapter 4: Complex Projective Spaces

Chapter 5: Outline Of Quantum Mechanics

Chapter 6: Coherent States And Group Actions

Chapter 7: The Stellar Representation

Chapter 8: The Space Of Density Matrices

Chapter 9: Purification Of Mixed Quantum States

Chapter 10: Quantum Operations

Chapter 11: Duality: Maps Versus States

Chapter 12: Discrete Structures In Hilbert Space

Chapter 13: Density Matrices And Entropies

Chapter 14: Distinguishability Measures

Chapter 15: Monotone Metrics And Measures

Chapter 16: Quantum Entanglement

Chapter 17: Multipartite Entanglement

About the Authors

Ingemar Bengtsson is a professor of physics at Stockholms Universitet. After gaining a Ph.D. in Theoretical Physics from the Göteborgs universitet (1984), he held post-doctoral positions at CERN, Geneva, and the Imperial College of Science, Technology and Medicine, University of London. He returned to Göteborg in 1988 as a research assistant at Chalmers tekniska högskola, before taking up a position as Lecturer in Physics at Stockholms Universitet in 1993. He 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. His research interests are related to geometry, in the forms of classical general relativity and quantum theory.

Karol Życzkowski is a professor at the Institute of Physics, Uniwersytet Jagiellonski, Kraków, Poland, and also at the Center for Theoretical Physics, Polska Akademia Nauk, Warsaw. He gained his Ph.D. and habilitation in theoretical physics at Uniwersytet Jagiellonski, and has followed this with a Humboldt Fellowship in Essen, a Fulbright Fellowship at the University of Maryland, College Park, and a visiting research position at the Perimeter Institute for Theoretical Physics, Ontario. He has been docent at the Academy of Sciences since 1999 and full professor at Uniwersytet Jagiellonski since 2004. Professor Życzkowski is a member of the Polish Physical Society and Academia Europaea. He works on quantum information, dynamical systems and chaos, quantum and statistical physics, applied mathematics, and theory of voting.