On realizing Lov\'asz-optimum orthogonal representation in the real Hilbert space

TL;DR

This paper proves that Lovász-optimum orthogonal representations of exclusivity graphs in complex Hilbert spaces can always be realized in real Hilbert spaces of appropriate dimension, advancing understanding of quantum contextuality.

Contribution

It demonstrates that Lovász-optimum orthogonal representations in complex spaces have real-space counterparts, completing the proof for their realizability in real Hilbert spaces.

Findings

Lovász-optimum representations in complex spaces can be realized in real spaces

The dimension of the real space needed is (2d-1) for a complex space of dimension d

This result completes the proof of real representability for all exclusivity graphs

Abstract

Quantum contextuality is usually revealed by the non-contextual inequality, which can always be associated with an exclusivity graph. The quantum upper bound of the inequality is nothing but the Lov\'asz number of the graph. In this work, we show that if there is a Lov\'asz-optimum orthogonal representation realized in the $d$-dimensional complex Hilbert space, then there always exists a corresponding Lov\'asz-optimum orthogonal representation in the $(2d-1)$-dimensional real Hilbert space. This in turn completes the proof that the Lov\'asz-optimum orthogonal representation for any exclusivity graph can always be realized in the real Hilbert space of suitable dimension.