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

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### Contents

Convexity colours and statistics | 1 |

Geometry of probability distributions | 29 |

Much ado about spheres | 63 |

Complex projective spaces | 103 |

Outline of quantum mechanics | 138 |

Coherent states and group actions | 165 |

The stellar representation | 191 |

The space of density matrices | 219 |

maps versus states | 296 |

Discrete structures in Hilbert space | 313 |

Density matrices and entropies | 355 |

Distinguishability measures | 386 |

Monotone metrics and measures | 402 |

Quantum entanglement | 433 |

Multipartite entanglement | 493 |

Epilogue | 544 |

### Other editions - View all

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

### Common terms and phrases

3-sphere affine algebra arbitrary basis bipartite bistochastic bundle Bures classical coherent completely positive compute convex set coordinates corresponding curve defined definition denote density matrices diagonal dimension dimensional distance dual dynamical matrix eigenvalues elements equal equation fibre Figure finite Fubini–Study generalised geodesic geometry given Hence Hermitian Hilbert space Hilbert–Schmidt Horodecki inequality invariant Kähler linear majorisation manifold maximally entangled maximally mixed measure monotone Neumann entropy normalised observe obtain octant orbit orthogonal partial trace phase space Phys plane polynomial polytope positive maps POVM probability distribution Problem projective pure quantum mechanics qubits random relative entropy Rényi entropies rotation Schmidt Schmidt decomposition Section separable Shannon entropy simplex SLOCC sphere stochastic subgroup submanifold subspace subsystems symmetric symplectic tangent tensor theorem theory transformations two-qubit unitary matrix unitary operator vector space von Neumann entropy zero