dorsal/arxiv
View SchemaDeduction, Ordering, and Operations in Quantum Logic
| Authors | Norman D. Megill, Mladen Pavicic |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0108074 |
| URL | https://arxiv.org/abs/quant-ph/0108074 |
| Journal | Found. Phys. 32 (2002) 357-378 |
Abstract
We show that in quantum logic of closed subspaces of Hilbert space one cannot substitute quantum operations for classical (standard Hilbert space) ones and treat them as primitive operations. We consider two possible ways of such a substitution and arrive at operation algebras that are not lattices what proves the claim. We devise algorithms and programs which write down any two-variable expression in an orthomodular lattice by means of classical and quantum operations in an identical form. Our results show that lattice structure and classical operations uniquely determine quantum logic underlying Hilbert space. As a consequence of our result, recent proposals for a deduction theorem with quantum operations in an orthomodular lattice as well as a substitution of quantum operations for the usual standard Hilbert space ones in quantum logic prove to be misleading. Quantum computer quantum logic is also discussed.
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"abstract": "We show that in quantum logic of closed subspaces of Hilbert space one cannot\nsubstitute quantum operations for classical (standard Hilbert space) ones and\ntreat them as primitive operations. We consider two possible ways of such a\nsubstitution and arrive at operation algebras that are not lattices what proves\nthe claim. We devise algorithms and programs which write down any two-variable\nexpression in an orthomodular lattice by means of classical and quantum\noperations in an identical form. Our results show that lattice structure and\nclassical operations uniquely determine quantum logic underlying Hilbert space.\nAs a consequence of our result, recent proposals for a deduction theorem with\nquantum operations in an orthomodular lattice as well as a substitution of\nquantum operations for the usual standard Hilbert space ones in quantum logic\nprove to be misleading. Quantum computer quantum logic is also discussed.",
"arxiv_id": "quant-ph/0108074",
"authors": [
"Norman D. Megill",
"Mladen Pavicic"
],
"categories": [
"quant-ph",
"math-ph",
"math.LO",
"math.MP",
"math.QA"
],
"journal_ref": "Found. Phys. 32 (2002) 357-378",
"title": "Deduction, Ordering, and Operations in Quantum Logic",
"url": "https://arxiv.org/abs/quant-ph/0108074"
},
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