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Equations, States, and Lattices of Infinite-Dimensional Hilbert Spaces

We provide several new results on quantum state space, on lattice of subspaces of an infinite dimensional Hilbert space, and on infinite dimensional Hilbert space equations as well as on connections between them. In particular we obtain an n-variable generalized orthoarguesian equation which holds in any infinite dimensional Hilbert space. Then we strengthen Godowski's result by showing that in an ortholattice on which strong states are defined Godowski's equations as well as the orthomodularity hold. We also prove that all 6- and 4-variable orthoarguesian equations presented in the literature can be reduced to new 4- and 3-variable ones, respectively and that Mayet's examples follow from Godowski's equations. To make a breakthrough in testing these massive equations we designed several novel algorithms for generating Greechie diagrams with an arbitrary number of blocks and atoms (currently testing with up to 50) and for automated checking of equations on them. A way of obtaining complex infinite dimensional Hilbert space from the Hilbert lattice equipped with several additional conditions and without invoking the notion of state is presented. Possible repercussions of the results to quantum computing problems are discussed.

Hilbert space; Hilbert lattice; orthoarguesian property; strong state; quantum logic; quantum computation; Godowski equations; orthomodular lattice

Our original title "Equations and State and Lattice Properties That Hold in Infinite Dimensional Hilbert Space, " which appears in the ArXiv files has been eventually changed into "Equations, States, and Lattices of Infinite-Dimensional Hilbert Spaces" by the editor of the International Journal of Theoretical Physics.