Vector generation of contextual sets

Abstract

As quantum contextuality proves to be a necessary resource for universal quantum computation, we present a general method for vector generation of Kochen-Specker (KS) contextual sets in the form of hypergraphs. The method supersedes all three previous methods: (i) fortuitous discoveries of smallest KS sets, (ii) exhaustive upward hypergraph-generation of sets, and (iii) random downward generation of sets from fortuitously obtained big master sets. In contrast to previous works, we can generate master sets which contain all possible KS sets starting with nothing but a few simple vector components. From them we can readily generate all KS sets obtained in the last half a century and any specified new KS sets. Herewith we can generate sufficiently large sets as well as sets with definite required features and structures to enable varieties of different implementations in quantum computation and communication.

Quantum Contextuality

“…The former representation in a complex space has been implemented in [10]. The complex vectors used for the implementation promptly give all possible smallest MMPHs since their master MMPH is fairly small [18,20,27]. However, in the real space the 332-1408 class generated by {0, ±1} components is huge and already at the time of its generation [16,18] we were well aware that then obtained 34-16 critical might not be the smallest.…”

Quantum Contextuality

“…Peres himself tried to obtain smaller sets via a computer program but failed [45]. It took three years until Kernaghan obtain one smaller set [46] and two more years until Cabello, Estebaranz, and García-Alcaine obtained a second one [47]; a straightforward translation of Peres' 24-24 set into an MMP 24-24 hypergraph string [48] immediately enables one to obtain all 1,233 KS MMP subhypergraphs on any laptop in seconds [49]; a graphical representation of the MMP 24-24 [48] even enables one to obtain desired subhypergraphs by hand hand in minutes [50];…”