Geometry of quantum states. An introduction to quantum entanglement.

*(English)*Zbl 1146.81004
Cambridge: Cambridge University Press (ISBN 0-521-81451-0/hbk; 978-0-511-53504-8/ebook). xii, 466 p. (2006).

This book is a serious and successful attempt to explain the tools used or useful to investigate structures of quantum states especially with respect to quantum information theory. Mathematically, these tools belong mainly to affine geometry as well as differential geometry. The tools are motivated, introduced, explained, and interrelated in a precise and animating language. Well known theorems are spelled out, carefully mentioning the origin and development. For example any reader will know the theorem characterizing the pure decompositions of a density operator, but only a few of them will know that this was published already in 1936 by Erwin Schrödinger. Proofs are often omitted or merely sketched. The authors expect the reader to have some knowledge to complete the proofs or to find them elsewhere. Not interrupted by lengthy proofs, this text gives on 416 pages of the main part a rich and lucid picture of the geometry of quantum states. Facts are, whenever possible, made graphic by instructive figures. Each chapter ends with a collection of interesting problems the solutions of which are given in Appendix 4. Appendices 1 and 2 contain basic notions on differential geometry and group theory. Appendix 3 poses some further exercises.

The 15 main chapters are: “Convexity, colors and statistics”; “Geometry of probability distributions” including majorization, classical measures of information, Fisher-Rao metric, generalized entropies; “Much ado about spheres” including Kähler and symplectic manifolds, Hopf fibration of the 3-sphere, fibre bundles and connections: “Complex projective geometry” including stars, spinors and the Fubini-Study metric; “Quantum mechanics”; “Coherent states and group actions”; “Stellar representations” including the Husimi function, Wehrl entropy and Lieb conjecture, Monge distance; “The space of density matrices” containing Hilbert-Schmidt and trace norm, convexity; “Purification of mixed quantum states” containing the Schmidt decomposition, the Hilbert-Schmidt bundle; “Quantum operations” containing POV-measures, complete positivity, the Kraus representation, unital and bistochastic maps; “Duality: maps versus states” containing decomposable positive maps, dual cones and super-positive maps, the Jamiolkowski isomorphism; “Density matrices and entropies” includig von Neumann entropy, quantum relative entropy and other constructs, majorization of states and possible state changes; “Distinguishability of states” containing classical and quantum state discrimination, fidelity and statistical distance; “Monotone metrics and measures” containing product measures and flag manifolds, Hilbert-Schmidt-, Bures- and induced measures, random density operators and random operations. After these preparations quantum entanglement is introduced and discussed in chapter 15, “Quantum entanglement”, further containing separability criteria, the geometry of the subset of disentangled (here also called separable) states, the entanglement measures, concurrences, Wootters formula which is derived and dicussed. An Epilogue closes the main part.

This very rich and readable text is based on about 675 citations. It can be best recommended as a guide to learn or to round up the knowledge about quantum states, entanglement and quantum information theory in general.

The 15 main chapters are: “Convexity, colors and statistics”; “Geometry of probability distributions” including majorization, classical measures of information, Fisher-Rao metric, generalized entropies; “Much ado about spheres” including Kähler and symplectic manifolds, Hopf fibration of the 3-sphere, fibre bundles and connections: “Complex projective geometry” including stars, spinors and the Fubini-Study metric; “Quantum mechanics”; “Coherent states and group actions”; “Stellar representations” including the Husimi function, Wehrl entropy and Lieb conjecture, Monge distance; “The space of density matrices” containing Hilbert-Schmidt and trace norm, convexity; “Purification of mixed quantum states” containing the Schmidt decomposition, the Hilbert-Schmidt bundle; “Quantum operations” containing POV-measures, complete positivity, the Kraus representation, unital and bistochastic maps; “Duality: maps versus states” containing decomposable positive maps, dual cones and super-positive maps, the Jamiolkowski isomorphism; “Density matrices and entropies” includig von Neumann entropy, quantum relative entropy and other constructs, majorization of states and possible state changes; “Distinguishability of states” containing classical and quantum state discrimination, fidelity and statistical distance; “Monotone metrics and measures” containing product measures and flag manifolds, Hilbert-Schmidt-, Bures- and induced measures, random density operators and random operations. After these preparations quantum entanglement is introduced and discussed in chapter 15, “Quantum entanglement”, further containing separability criteria, the geometry of the subset of disentangled (here also called separable) states, the entanglement measures, concurrences, Wootters formula which is derived and dicussed. An Epilogue closes the main part.

This very rich and readable text is based on about 675 citations. It can be best recommended as a guide to learn or to round up the knowledge about quantum states, entanglement and quantum information theory in general.

Reviewer: K.-E. Hellwig (Berlin)

##### MSC:

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

81P68 | Quantum computation |

81-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory |

81P05 | General and philosophical questions in quantum theory |

81P15 | Quantum measurement theory, state operations, state preparations |

94A17 | Measures of information, entropy |