Geometry of Quantum States:
  An Introduction to Quantum Entanglement

Ingemar Bengtsson and Karol Zyczkowski

second revised edition, Cambridge University Press, Cambridge, 2017

CONTENTS  of the second edition - entirely new fragments in green


  1. Convexity, colours and statistics
1.1  Convex sets                               
1.2  High dimensional geometry
1.3  Colour theory
1.4  What is "distance'' ?
1.5  Probability and statistics

  2. Geometry of probability distributions
2.1  Majorization and partial order
2.2  Shannon entropy
2.3  Relative entropy
2.4  Continuous distributions and measures
2.5  Statistical geometry and the Fisher--Rao metric
2.6  Classical ensembles
2.7  Generalized entropies

3. Much ado about spheres
3.1  Spheres
3.2  Parallel transport and statistical geometry
3.3  Complex, Hermitian, and Kaehler manifolds
3.4  Symplectic manifolds
3.5  The Hopf fibration of the 3-sphere
3.6  Fibre bundles and their connections
3.7  The 3-sphere as a group
3.8  Cosets and all that
4.  Complex projective spaces
4.1  From art to mathematics
4.2  Complex projective geometry
4.3  Complex curves, quadrics and the Segre embedding
4.4  Stars, spinors, and complex curves
4.5  The Fubini-Study metric
4.6  CPn illustrated
4.7  Symplectic geometry and the Fubini--Study measure
4.8  Fibre bundle aspects
4.9  Grassmannians and flag manifolds

 5. Outline of quantum mechanics
5.1  Quantum mechanics
5.2  Qubits and Bloch spheres
5.3  The statistical and the Fubini-Study distances
5.4  A real look at quantum dynamics
5.5  Time reversals
5.6  Classical & quantum states: a unified approach
5.7  Gleason and Kochen–Specker

  6. Coherent states and group actions
6.1  Canonical coherent states
6.2  Quasi-probability distributions on the plane
6.3  Bloch coherent states
6.4  From complex curves to SU(K) coherent states
6.5  SU(3) coherent states

  7. The stellar representation

7.1  The stellar representation in quantum mechanics
7.2  Orbits and coherent states
7.3  The Husimi function
7.4  Wehrl entropy and the Lieb conjecture
7.5  Generalised Wehrl entropies
7.6  Random pure states
7.7  From the transport problem to the Monge distance

  8. The space of density matrices
8.1  Hilbert--Schmidt space and positive operators
8.2  The set of mixed states
8.3  Unitary transformations
8.4  The space of density matrices as a convex set
8.5  Stratification
8.6  Projections and cross-sections
8.6  An algebraic afterthought
8.7  Summary

  9.  Purification of mixed quantum states

9.1  Tensor products and state reduction
9.2  The Schmidt decomposition
9.3  State purification  & the Hilbert-Schmidt bundle
9.4  A first look at the Bures metric
9.5  Bures geometry for  N=2
9.6  Further properties of the Bures metric

 10. Quantum operations
10.1  Measurements and POVMs
10.2  Algebraic detour: matrix reshaping and reshuffling
10.3  Positive and completely positive maps
10.4  Environmental representations
10.5  Some spectral properties
10.6  Unital & bistochastic maps
10.7  One qubit maps

 11.  Duality: maps versus states
11.1  Positive & decomposable maps
11.2  Dual cones and super-positive maps
11.3  Jamiolkowski isomorphism
11.4  Quantum maps and quantum states

12.  Discrete structures in Hilbert spacesee preprint arXiv: 1701.07902
        12.1 Unitary operator bases and the Heisenberg groups
        12.2 Prime, composite and prime power dimensions
        12.3 More unitary operator bases
        12.4 Mutually unbiased bases
        12.5 Finite geometries and discrete Wigner functions
        12.6 Clifford groups and stabilizer states
        12.7 Some designs
        12.8 SICs

 13.  Density matrices and entropies
13.1  Ordering operators
13.2  Von Neumann entropy
13.3  Quantum relative entropy
13.4  Other entropies
13.5  Majorization of density matrices
13.6  Proof of the Lieb conjecture
13.7  Entropy dynamics

 14. Distinguishability measures
14.1  Classical distinguishability measures
14.2  Quantum distinguishability measures
14.3  Fidelity and statistical distance

 15.  Monotone metrics and measures
15.1  Monotone metrics
15.2  Product measures and flag manifolds
15.3  Hilbert-Schmidt measure
15.4  Bures measure
15.5  Induced measures
15.6  Random density matrices
15.7  Random operations
15.8 Concentration of measure

  16.  Quantum entanglement  (draft)
 16.1  Introducing entanglement
 16.2  Two qubit pure states: entanglement illustrated
 16.3 Maximally entangled states
 16.4 Pure states of a bipartite system
 16.5 A first look at entangled mixed states
 16.6 Separability criteria
 16.7 Geometry of the set of separable states
 16.8 Entanglement measures
 16.9 Two-qubit mixed states

  17.  Multipartite entanglement - see preprint arXiv: 1612.07747
17.1 How much is three larger than two?
17.2 Botany of states
17.3 Permutation symmetric states
17.4 Invariant theory and quantum states
17.5 Monogamy relations and global multipartite entanglement
17.6 Local spectra and the momentum map
17.7 AME states and error-correcting codes
17.8 Entanglement in quantum spin systems


Appendix A.  Basic notions of differential geometry 
Appendix B.  Basic notions of group theory
Appendix C.  Geometry do it yourself
Appendix D. 
Hints & answers to the exercises

Bibliography  & Index


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