Colouring an Orthogonality Graph

TL;DR

This paper solves a graph colouring problem from quantum information theory, showing that classical strategies can succeed without entanglement only for vectors of length up to 3, clarifying the boundary between classical and quantum success.

Contribution

The paper precisely characterizes when classical strategies can succeed in a quantum information task by solving the associated graph colouring problem.

Findings

Classical success is possible only for vectors of length up to 3.

Quantum entanglement allows success for all lengths.

The graph colouring problem is fully resolved for this scenario.

Abstract

We deal with a graph colouring problem that arises in quantum information theory. Alice and Bob are each given a $\pm1$-vector of length $k$, and are to respond with $k$ bits. Their responses must be equal if they are given equal inputs, and distinct if they are given orthogonal inputs; however, they are not allowed to communicate any information about their inputs. They can always succeed using quantum entanglement, but their ability to succeed using only classical physics is equivalent to a graph colouring problem. We resolve the graph colouring problem, thus determining that they can succeed without entanglement exactly when $k\leq3$.