Hilbert Lattice Equations

Abstract

Abstract. There are five known classes of lattice equations that hold in every infinite dimensional Hilbert space underlying quantum systems: generalised orthoarguesian, Mayet's EA, Godowski, Mayet-Godowski, and Mayet's E equations. We obtain a result which opens a possibility that the first two classes coincide. We devise new algorithms to generate Mayet-Godowski equations that allow us to prove that the fourth class properly includes the third. An open problem related to the last class is answered. Finally, we show some new results on the Godowski lattices characterising the third class of equations. Mathematics Subject Classification (2000). Primary 46C15; Secondary 06B20.

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.…”

“…The cycles themselves will allow us to generate new lattice equations following the procedure developed in 19,21,67 , but they do not automatically follow possible geometrical symmetries of the hypergraphs. In the 36-36 case they do, but, e.g., they do not exhibit the left right symmetry of the 35-35 lattice shown in Fig.…”

Kochen–Specker Sets and Generalized Orthoarguesian Equations

“…The left figure shows the blocks we dropped from Fig. 2, and the right one is given in the representation we previously used to show violations of 3OA through 6OA at lattices presented in [2,6,20] with the maximal loop (tetrakaidecagon, 14-gon) it contains.…”

“…Since 1975, additional equations that it satisfies have been discovered. Among these, the only ones known that are directly related to the vector space of the underlying Hilbert space (i.e., excluding those that are related to states introduced on the lattice) are the generalized orthoarguesian equations (nOA, n ≥ 3) [2]. Thus, these equations are an essential tool for analyzing lattices conjectured to represent particular experimental setups.…”

Logical aspects of quantum structures

“…He showed that this equation is valid in C(H) and not in oml. Many refinements of this ortho-Arguesian identity have been found [55,56] providing other equations valid in C(H) and not in oml. A further source of equations valid in C(H) and not in oml is provided by the fact that C(H) has an ample supply of well-behaved states [27,53,54].…”