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
Preface
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 space - see 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
Epilogue
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
