An elementary introduction to the geometry of quantum states with a picture book

1 Introduction

1.1 The geometry of quantum states

• 3 Basic geometry of Quantum states
  
  
  • 3.1 Choosing coordinates
  
  
  • 3.2 Most of the unit sphere does not represent states
  
  
  • 3.3 Inversion asymmetry
  
  
  • 3.4 The inscribed sphere
  
  
  
  


• 4 Cross sections
  
  
  • 4.1 Cross sections that are N-1 simplexes
  
  
  • 4.2 Cross sections that are balls
  
  
  • 4.3 Cross sections that are polyhedra and hyper-octahedra
  
  
  • 4.4 2D cross sections in the Pauli basis
  
  
  
  


• 5 The radius function
  
  
  • 5.1 The radius function is N-Lifshitz
  
  
  
  


• 6 A tiny ball in most directions
  
  
  • 6.1 Application of random matrix theory
  
  
  • 6.2 Directions associated with states with substantial purity are rare
  
  
  
  


• 7 Separable and entangled states
  
  
  • 7.1 Why separability is hard
  
  
  • 7.2 Completely separable simplex: Classical bits
  
  
  • 7.3 Entangled pure states
  
  
  • 7.4 Two types of entangled states
  
  
  • 7.5 The largest ball of bi-partite separable states
  
  
  • 7.6 Entanglement witnesses
  
  
  • 7.7 Entangled states and witnesses near the Gurvits-Barnum ball
  
  
  • 7.8 A Clifford ball of separable states
  
  
  
  


• A The average purity of quantum states


• B The N dimensional unit cube is almost a ball


• References