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

• URL: http://arxiv.org/abs/2511.17223v1
• Date: Fri, 21 Nov 2025 13:10:09 GMT
• Title: Phase-adjusted realification of a $\mathbb{C}^3$ Kochen-Specker configuration into $\mathbb{R}^6$
• Authors: Andrei Khrennikov, Karl Svozil,
• Abstract summary: A phase-adjusted realification procedure embeds any finite set of rays in $mathbbC3$ into $mathbbR6$.<n>We consider the 165 projectively distinct rays used in a $mathbbC3$ Kochen-Specker configuration obtained from mutually unbiased bases.<n>Because the original 3-element contexts are no longer maximal in $mathbbR6$, the embedded configuration admits two-valued states even though its realisation with maximal contexts in $mathbbC3$ is Kochen-Specker uncolourable
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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