dorsal/arxiv
View SchemaEquations, States, and Lattices of Infinite-Dimensional Hilbert Spaces
| Authors | Norman D. Megill, Mladen Pavicic |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0009038 |
| URL | https://arxiv.org/abs/quant-ph/0009038 |
| Journal | International Journal of Theoretical Physics, 39, 2337-2379 (2000) |
Abstract
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.
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"abstract": "We provide several new results on quantum state space, on lattice of\nsubspaces of an infinite dimensional Hilbert space, and on infinite dimensional\nHilbert space equations as well as on connections between them. In particular\nwe obtain an n-variable generalized orthoarguesian equation which holds in any\ninfinite dimensional Hilbert space. Then we strengthen Godowski\u0027s result by\nshowing that in an ortholattice on which strong states are defined Godowski\u0027s\nequations as well as the orthomodularity hold. We also prove that all 6- and\n4-variable orthoarguesian equations presented in the literature can be reduced\nto new 4- and 3-variable ones, respectively and that Mayet\u0027s examples follow\nfrom Godowski\u0027s equations. To make a breakthrough in testing these massive\nequations we designed several novel algorithms for generating Greechie diagrams\nwith an arbitrary number of blocks and atoms (currently testing with up to 50)\nand for automated checking of equations on them. A way of obtaining complex\ninfinite dimensional Hilbert space from the Hilbert lattice equipped with\nseveral additional conditions and without invoking the notion of state is\npresented. Possible repercussions of the results to quantum computing problems\nare discussed.",
"arxiv_id": "quant-ph/0009038",
"authors": [
"Norman D. Megill",
"Mladen Pavicic"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP",
"math.QA"
],
"journal_ref": "International Journal of Theoretical Physics, 39, 2337-2379 (2000)",
"title": "Equations, States, and Lattices of Infinite-Dimensional Hilbert Spaces",
"url": "https://arxiv.org/abs/quant-ph/0009038"
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