Small Quantum Structures with Small State Spaces

Abstract

We summarize and extend results about "small" quantum structures with small dimensions of state spaces. These constructions have contributed to the theory of orthomodular lattices. More general quantum structures (orthomodular posets, orthoalgebras, and effect algebras) admit sometimes simplifications, but there are problems where no progress has been achieved.

Graph approach to quantum systems

“…II.8) using lattices that do not admit strong sets of states. 19,21 Based on all that together with several previous results based on lattices admitting only one state, 32,33,71,72 in Sec. II we formulated the following theorem: Theorem II.13 [Semi-quantum lattice algorithms] There exist algorithms that generate finite sequences of OMLs that admit superposition, real-valued states, and a vector state given by Eq.…”

Existence of states on quantum structures

“…It has been described by Riečanová in [11]. Details about these (and other related) examples can be found in [10].…”

Modelling of Uncertainty and Bi–Variable Maps

“…It is known, that there exist orthomodular lattices without any probability measure [12,21] and also such that have only one probability measure [21]. If there are at least two different probability measures defined on an OML L, then the set of probability measures defined on L is obviously uncountable since any convex combination of probability measures is a probability measure as well.…”