Quantum Contextuality

Abstract

Quantum contextual sets have been recognized as resources for universal quantum computation, quantum steering and quantum communication. Therefore, we focus on engineering the sets that support those resources and on determining their structures and properties. Such engineering and subsequent implementation rely on discrimination between statistics of measurement data of quantum states and those of their classical counterparts. The discriminators considered are inequalities defined for hypergraphs whose structure and generation are determined by their basic properties. The generation is inherently random but with the predetermined quantum probabilities of obtainable data. Two kinds of statistics of the data are defined for the hypergraphs and six kinds of inequalities. One kind of statistics, often applied in the literature, turn out to be inappropriate and two kinds of inequalities turn out not to be noncontextuality inequalities. Results are obtained by making use of universal automated algorithms which generate hypergraphs with both odd and even numbers of hyperedges in any odd and even dimensional space – in this paper, from the smallest contextual set with just three hyperedges and three vertices to arbitrarily many contextual sets in up to 8-dimensional spaces. Higher dimensions are computationally demanding although feasible.

Non-Kochen–Specker Contextuality

“…For = , an NBMMPH reduces to a KS contextuality set, i.e., to a set satisfying the Kochen–Specker theorem [ 32 , 34 , 45 , 46 ].…”

“…M1 consists of dropping vertices contained in single hyperedges (multiplicity ) [ 34 ] of either NBMMPHs or BMMPHs and a possible subsequent stripping of their hyperedges. The obtained smaller MMPHs are often non-KS, although never KS.…”

“…Howard, Wallman, Veitech, and Emerson’s four-dim 30-108 non-KS NBMMPH, shown in Figure 1 b, which was obtained from the set of stabilizer states was used to prove that the underlying contextuality is essential for quantum computation. We discuss its filled 232-108 MMPH and its critical 24-71 MMPH in [ 34 ].…”

“…There are three classes of contextual sets elaborated on in the literature which are not of the more common kind of Kochen–Specker (KS) sets [ 32 , 33 , 34 ] and for which we provide a hypergraph generalization in this paper.…”

“…The second class consists of hypergraphs built by multiples of mutually orthogonal vectors where at least one of the multiples contains less than n vectors, where n is the dimension of space in which a hypergraph resides [ 4 , 34 , 41 ].…”