Ravishankar Ramanathan, Pawel Horodecki
Contextuality of non-perfect orthogonality graphs is studied and shown to occur even in cases when the Lovasz and independence numbers are equal. It is proven that for given dimension (i) contextuality of a quantum state is entirely determined by its spectrum, hence pure and maximally mixed states represent the two extremes of contextual behavior and (ii) state-independent contextuality (S-IC) is equivalent to contextuality of the maximally mixed state. A necessary and sufficient condition for a graph to be S-IC is provided in terms of the fractional chromatic number $\chi_f (G)$ and is shown to be stronger than its direct analog in terms of the (standard) chromatic number $\chi(G)$. The completion paradigm is then studied in which extra projections required to physically realize a complete measurement are shown to constitute a "cloud of contextuality" which may render the full set of projectors S-IC even if the original set of projectors was not. On that new level, the intriguing interplay between the $\chi_f (G)$ and $\chi(G)$ conditions is shown to lead to a striking property of the well-known Kneser graphs and subgraphs in regard to their quantum realization.
View original:
http://arxiv.org/abs/1212.5933
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