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

TL;DR

This paper presents a phase-adjusted realification method that faithfully embeds complex three-dimensional Kochen-Specker configurations into real space, preserving orthogonality relations while revealing new properties of the embedded structure.

Contribution

It introduces a novel phase-adjusted realification procedure for embedding complex configurations into real space, enabling faithful orthogonal representations of Kochen-Specker sets.

Findings

Successfully embeds 165 rays from a complex Kochen-Specker configuration into R^6.

Shows the embedded configuration admits two-valued states despite original uncolourability.

Provides explicit list of rays and their realified images in R^6.

Abstract

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