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

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.