Geometry of Quantum States

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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. Discrete structures in Hilbert space; 13. Density matrices and entropies; 14. Distinguishability measures; 15. Monotone metrics and measures; 16. Quantum entanglement; 17. Multipartite entanglement; Appendix 1. Basic notions of differential geometry; Appendix 2. Basic notions of group theory; Appendix 3. Geometry - do it yourself; Appendix 4. Hints and answers to the exercises; Bibliography; Index.

This new edition describes the space of quantum states and the theory of quantum entanglement from a geometric perspective.

Ingemar Bengtsson is a professor of physics at Stockholms Universitet. After gaining a Ph.D. in Theoretical Physics from the Goeteborgs 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 Goeteborg in 1988 as a research assistant at Chalmers tekniska hoegskola, 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 Zyczkowski is a professor at the Institute of Physics, Uniwersytet Jagiellonski, Krakow, 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 Zyczkowski 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.

'True story: A few years ago my daughter took a break from her
usual question, 'Dad, what is your favourite colour?' and asked
instead, 'What is your favourite shape?' I was floored! 'What a
wonderful question; my favourite shape is Hilbert space!' 'What
does it look like?' she asked. My answer: 'I don't know! But every
day when I go to work, that's what I think about.' What I was
speaking of, of course, is the geometry of quantum-state space. It
is as much a mystery today as it was those years ago, and maybe
more so as we learn to focus on its most key and mysterious
features. This book, the worn first-edition of which I've had on my
shelf for 11 years, is the indispensable companion for anyone's
journey into that exotic terrain. Beyond all else, I am thrilled
about the inclusion of two new chapters in the new edition, one of
which I believe goes to the very heart of the meaning of quantum
theory.' Christopher A. Fuchs, University of Massachusetts,
Boston

'The quantum world is full of surprises as is the mathematical
theory that describes it. Bengtsson and Zyczkowski prove to be
expert guides to the deep mathematical structure that underpins
quantum information science. Key concepts such as multipartite
entanglement and quantum contextuality are discussed with
extraordinary clarity. A particular feature of this new edition is
the treatment of SIC generalised measurements and the curious
bridge they make between quantum physics and number theory.' Gerard
J. Milburn, University of Queensland

Praise for the first edition: '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, editor of Open Systems and
Information Dynamics

Praise for the first edition: '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

Advance praise: 'True story: A few years ago my daughter took a
break from her usual question, 'Dad, what is your favourite
colour?' and asked instead, 'What is your favourite shape?' I was
floored! 'What a wonderful question; my favourite shape is Hilbert
space!' 'What does it look like?' she asked. My answer: 'I don't
know! But every day when I go to work, that's what I think about.'
What I was speaking of, of course, is the geometry of quantum-state
space. It is as much a mystery today as it was those years ago, and
maybe more so as we learn to focus on its most key and mysterious
features. This book, the worn first-edition of which I've had on my
shelf for 11 years, is the indispensable companion for anyone's
journey into that exotic terrain. Beyond all else, I am thrilled
about the inclusion of two new chapters in the new edition, one of
which I believe goes to the very heart of the meaning of quantum
theory.' Christopher A. Fuchs, University of Massachusetts,
Boston

Advance praise: 'The quantum world is full of surprises as is the
mathematical theory that describes it. Bengtsson and Zyczkowski
prove to be expert guides to the deep mathematical structure that
underpins quantum information science. Key concepts such as
multipartite entanglement and quantum contextuality are discussed
with extraordinary clarity. A particular feature of this new
edition is the treatment of SIC generalised measurements and the
curious bridge they make between quantum physics and number
theory.' Gerard J. Milburn, University of Queensland

Praise for the first edition: '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, editor of Open Systems and
Information Dynamics

Praise for the first edition: '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

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