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

We describe a phase-adjusted realification procedure that embeds any finite set of rays in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msup> <a:mi mathvariant="double-struck">C</a:mi> <a:mn>3</a:mn> </a:msup> </a:math> into <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:msup> <c:mi mathvariant="double-struck">R</c:mi> <c:mn>6</c:mn> </c:msup> </c:math> . By assigning an appropriate phase to each ray before applying the standard coordinatewise map, we can arrange that two rays are orthogonal in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:msup> <e:mi mathvariant="double-struck">C</e:mi> <e:mn>3</e:mn> </e:msup> </e:math> if and only if their images are orthogonal in <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:msup> <g:mi mathvariant="double-struck">R</g:mi> <g:mn>6</g:mn> </g:msup> </g:math> , 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 <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:msup> <i:mi mathvariant="double-struck">C</i:mi> <i:mn>3</i:mn> </i:msup> </i:math> Kochen-Specker configuration obtained from mutually unbiased bases, list these 165 rays explicitly in <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:msup> <k:mi mathvariant="double-struck">C</k:mi> <k:mn>3</k:mn> </k:msup> </k:math> , and give for each of them its image in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mn>6</m:mn> </m:msup> </m:math> under the canonical realification map. We also note that, because the original three -element contexts are no longer maximal in <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:msup> <o:mi mathvariant="double-struck">R</o:mi> <o:mn>6</o:mn> </o:msup> </o:math> , the embedded configuration admits two-valued states even though its realization with maximal contexts in <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"> <q:msup> <q:mi mathvariant="double-struck">C</q:mi> <q:mn>3</q:mn> </q:msup> </q:math> is Kochen-Specker uncolorable.