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arXiv · 1805.08334

Spectral lower bounds for the quantum chromatic number of a graph

Pawel Wocjan·Clive Elphick
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Abstract

The quantum chromatic number, $\chi_q(G)$, of a graph $G$ was originally defined as the minimal number of colors necessary in a quantum protocol in which two provers that cannot communicate with each other but share an entangled state can convince an interrogator with certainty that they have a coloring of the graph. We use an equivalent purely combinatorial definition of $\chi_q(G)$ to prove that many spectral lower bounds for the chromatic number, $\chi(G)$, are also lower bounds for $\chi_q(G)$. This is achieved using techniques from linear algebra called pinching and twirling. We illustrate our results with some examples.

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Pawel Wocjan, Clive Elphick. 2018-05-22. Spectral lower bounds for the quantum chromatic number of a graph. https://arxiv.org/abs/1805.08334

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