Phase-adjusted realification of a $\mathbb{C}^3$ Kochen-Specker configuration into $\mathbb{R}^6$

Abstract

We describe a phase-adjusted realification procedure that embeds any finite set of rays in $\mathbb{C}^3$ into $\mathbb{R}^6$. By assigning an appropriate phase to each ray before applying the standard coordinate-wise map, we can arrange that two rays are orthogonal in $\mathbb{C}^3$ if and only if their images are orthogonal in $\mathbb{R}^6$, so the construction yields a faithful orthogonal representation of the original complex configuration. As a concrete example, we consider the 165 projectively distinct rays used in a $\mathbb{C}^3$ Kochen-Specker configuration obtained from mutually unbiased bases, list these 165 rays explicitly in $\mathbb{C}^3$, and give for each of them its image in $\mathbb{R}^6$ under the canonical realification map. We also note that, because the original 3-element contexts are no longer maximal in $\mathbb{R}^6$, the embedded configuration admits two-valued states even though its realisation with maximal contexts in $\mathbb{C}^3$ is Kochen-Specker uncolourable.