Faithful real embedding of a three-dimensional complex Kochen-Specker configuration

Abstract

We describe a phase-adjusted realification procedure that embeds any finite set of rays in ℂ³ into ℝ⁶. By assigning an appropriate phase to each ray before applying the standard coordinatewise map, we can arrange that two rays are orthogonal in ℂ³ if and only if their images are orthogonal in ℝ⁶, 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 ℂ³ Kochen-Specker configuration obtained from mutually unbiased bases, list these 165 rays explicitly in ℂ³, and give for each of them its image in ℝ⁶ under the canonical realification map. We also note that, because the original three -element contexts are no longer maximal in ℝ⁶, the embedded configuration admits two-valued states even though its realization with maximal contexts in ℂ³ is Kochen-Specker uncolorable.