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The quantum-to-classical graph homomorphism game

Michael Brannan, Priyanga Ganesan, Samuel J. Harris

Abstract

Motivated by non-local games and quantum coloring problems, we introduce a graph homomorphism game between quantum graphs and classical graphs. This game is naturally cast as a "quantum-classical game"--that is, a non-local game of two players involving quantum questions and classical answers. This game generalizes the graph homomorphism game between classical graphs. We show that winning strategies in the various quantum models for the game is an analogue of the notion of non-commutative graph homomorphisms due to D. Stahlke [44]. Moreover, we present a game algebra in this context that generalizes the game algebra for graph homomorphisms given by J.W. Helton, K. Meyer, V.I. Paulsen and M. Satriano [22]. We also demonstrate explicit quantum colorings of all quantum complete graphs, yielding the surprising fact that the algebra of the $4$-coloring game for a quantum graph is always non-trivial, extending a result of [22].

The quantum-to-classical graph homomorphism game

Abstract

Motivated by non-local games and quantum coloring problems, we introduce a graph homomorphism game between quantum graphs and classical graphs. This game is naturally cast as a "quantum-classical game"--that is, a non-local game of two players involving quantum questions and classical answers. This game generalizes the graph homomorphism game between classical graphs. We show that winning strategies in the various quantum models for the game is an analogue of the notion of non-commutative graph homomorphisms due to D. Stahlke [44]. Moreover, we present a game algebra in this context that generalizes the game algebra for graph homomorphisms given by J.W. Helton, K. Meyer, V.I. Paulsen and M. Satriano [22]. We also demonstrate explicit quantum colorings of all quantum complete graphs, yielding the surprising fact that the algebra of the -coloring game for a quantum graph is always non-trivial, extending a result of [22].

Paper Structure

This paper contains 6 sections, 38 theorems, 131 equations.

Table of Contents

  1. Quantum Input, Classical Output Correlations
  2. Synchronous Quantum Input--Classical Output Correlations
  3. Approximately finite-dimensional correlations
  4. The game for quantum-to-classical graph homomorphisms
  5. Coloring quantum graphs
  6. Quantum Complete Graphs and Algebraic Colorings

Key Result

Proposition \oldthetheorem

Let $\mathcal{H}$ be a Hilbert space, and let $\{Q_a\}_{a=1}^c$ be a POVM in $\mathcal{B}(\mathcal{H})$. Then there is a PVM $\{P_a\}_{a=1}^c$ in $M_{c+1}(\mathcal{B}(\mathcal{H}))$ such that, if $E_{11}$ is the first diagonal matrix unit in $M_{c+1}$, then $(E_{11} \otimes I_{\mathcal{H}})P_a(E_{11

Theorems & Definitions (87)

  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • ...and 77 more