## Geometry of Quantum States: An Introduction to Quantum EntanglementQuantum 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. |

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### Contents

1 | |

Outline of quantum mechanics | 135 |

Coherent states and group actions | 156 |

The stellar representation | 182 |

The space of density matrices | 209 |

Puriﬁcation of mixed quantum states | 233 |

Quantum operations | 251 |

maps versus states | 281 |

Distinguishability measures | 323 |

Monotone metrics and measures | 339 |

Quantum entanglement | 363 |

1 | ix |

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440 | |

446 | |

452 | |

### Other editions - View all

Geometry of Quantum States: An Introduction to Quantum Entanglement Ingemar Bengtsson,Karol Życzkowski No preview available - 2006 |

Geometry of Quantum States: An Introduction to Quantum Entanglement Ingemar Bengtsson,Karol Zyczkowski No preview available - 2006 |

Geometry of Quantum States: An Introduction to Quantum Entanglement Ingemar Bengtsson,Karol Zyczkowski No preview available - 2007 |

### Common terms and phrases

3-sphere afﬁne algebra arbitrary bistochastic Bloch ball bundle Bures classical coherent completely positive compute cone convex set coordinates corresponding CP map curve deﬁned deﬁnition denote density matrices diagonal difﬁcult dimension dimensional dynamical matrix eigenstates eigenvalues equal equation ﬁbre ﬁeld Figure ﬁnd ﬁnite ﬁrst ﬁxed ﬂat geodesic geometry given Hence Hermitian matrices Hilbert space Hilbert–Schmidt Horodecki Husimi function inequality inﬁnity invariant K¨ahler linear maximally mixed measure Monge distance monotone Neumann entropy observe obtain octant operator monotone orbit orthogonal parameter partial trace phase space plane points positive maps positive operators POVM probability distribution Problem projective space pure quantum mechanics qubit R´enyi random real numbers relative entropy reshufﬂing rotation Schmidt decomposition Section Shannon entropy sphere stochastic submanifold subspace symmetric tangent vector tensor theorem transformations unitary unitary matrix vector space von Neumann entropy Wehrl entropy Wigner function zero Zyczkowski