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Quantum independence and chromatic numbers

Chris Godsil, Mariia Sobchuk

Abstract

We construct a new graph on 120 vertices whose quantum and classical independence numbers are different. At the same time, we construct an infinite family of graphs whose quantum chromatic numbers are smaller than the classical chromatic numbers. Furthermore, we discover the relation to Kochen-Specker sets that characterizes quantum cocliques that are strictly bigger than classical ones. Finally, we prove that for graphs with independence number is two, quantum and classical independence numbers coincide.

Quantum independence and chromatic numbers

Abstract

We construct a new graph on 120 vertices whose quantum and classical independence numbers are different. At the same time, we construct an infinite family of graphs whose quantum chromatic numbers are smaller than the classical chromatic numbers. Furthermore, we discover the relation to Kochen-Specker sets that characterizes quantum cocliques that are strictly bigger than classical ones. Finally, we prove that for graphs with independence number is two, quantum and classical independence numbers coincide.
Paper Structure (10 sections, 16 theorems, 54 equations, 1 figure)

This paper contains 10 sections, 16 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Piovesan's example with $\alpha<\alpha_q$ is Cayley
  3. Construction of the isomorphism to Cayley graph
  4. New graph on 120 vertices with $\alpha <\alpha_q$
  5. Quantum independence number and Kochen-Specker sets
  6. Characterization of when $\alpha<\alpha_q$ using Kochen-Specker sets
  7. Quantum chromatic number
  8. A new infinite family of graphs with $\chi>\chi_q$
  9. Quantum inertia bound and its application
  10. Acknowledgements

Key Result

Proposition 2.1

$\alpha(G_p)<\alpha_q(G_p).$

Figures (1)

  • Figure 1: The graph $G_{13}$ from oddities

Theorems & Definitions (24)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 5.1
  • proof
  • ...and 14 more