Equivalences, Identities, Symmetric Differences, and Congruences in Orthomodular Lattices

It is shown that operations of equivalence cannot serve for building algebras which would induce orthomodular lattices as the operations of implication can. Several properties of equivalence operations have been investigated. Distributivity of equivalence terms and several other 3 variable expressions involving equivalence terms have been proved to hold in any orthomodular lattice. Symmetric differences have been shown to reduce to complements of equivalence terms. Some congruence relations related to equivalence operations and symmetric differences have been considered.Comment: 13 pages, 1 figure, 1 table; To be published in International Journal of Theoretical Physics, Vol. 42, No. 12 (2003); Web page: http://m3k.grad.hr/pavici

Vector Generation of Contextual Sets

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.Comment: 6 pages, 6 figures; Orally presented by M. Pavicic at Quantum Technology International Conference 2018 (QTech 2018), Paris, France, September 5-7, 2018; Extended version: https://www.youtube.com/watch?v=Bw2vItz5tr