State-Dependent Implication and Equivalence in Quantum Logic
Ideal occurrence of an event (projector) leads to the known change of a state (density operator) into (the Lüders state). It is shown that two events and give the same Lüders state if and only if the equivalence relation is valid. This relation determines equivalence classes. The set of them...
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| Language: | English |
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Wiley
2012-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2012/385341 |
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| author | Fedor Herbut |
| author_facet | Fedor Herbut |
| author_sort | Fedor Herbut |
| collection | DOAJ |
| description | Ideal occurrence of an event (projector) leads to the known change of a state (density operator) into (the Lüders state). It is shown that two events and give the same Lüders state if and only if the equivalence relation is valid. This relation determines equivalence classes. The set of them and each class, are studied in detail. It is proved that the range projector of the Lüders state can be evaluated as , where denotes the greatest lower bound, and is the null projector of . State-dependent implication extends absolute implication (which, in turn, determines the entire structure of quantum logic). and are investigated in a closely related way to mutual benefit. Inherent in the preorder is the state-dependent equivalence , defining equivalence classes in a given Boolean subalgebra. The quotient set, in which the classes are the elements, has itself a partially ordered structure, and so has each class. In a complete Boolean subalgebra, both structures are complete lattices. Physical meanings are discussed. |
| format | Article |
| id | doaj-art-027e3bf09aa94048887a1b753ce5a93c |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-027e3bf09aa94048887a1b753ce5a93c2025-02-03T05:53:16ZengWileyAdvances in Mathematical Physics1687-91201687-91392012-01-01201210.1155/2012/385341385341State-Dependent Implication and Equivalence in Quantum LogicFedor Herbut0Department of Mathematics, Physics, and Geological Sciences, Serbian Academy of Sciences and Arts, Knez Mihajlova 35, 11000 Belgrade, SerbiaIdeal occurrence of an event (projector) leads to the known change of a state (density operator) into (the Lüders state). It is shown that two events and give the same Lüders state if and only if the equivalence relation is valid. This relation determines equivalence classes. The set of them and each class, are studied in detail. It is proved that the range projector of the Lüders state can be evaluated as , where denotes the greatest lower bound, and is the null projector of . State-dependent implication extends absolute implication (which, in turn, determines the entire structure of quantum logic). and are investigated in a closely related way to mutual benefit. Inherent in the preorder is the state-dependent equivalence , defining equivalence classes in a given Boolean subalgebra. The quotient set, in which the classes are the elements, has itself a partially ordered structure, and so has each class. In a complete Boolean subalgebra, both structures are complete lattices. Physical meanings are discussed.http://dx.doi.org/10.1155/2012/385341 |
| spellingShingle | Fedor Herbut State-Dependent Implication and Equivalence in Quantum Logic Advances in Mathematical Physics |
| title | State-Dependent Implication and Equivalence in Quantum Logic |
| title_full | State-Dependent Implication and Equivalence in Quantum Logic |
| title_fullStr | State-Dependent Implication and Equivalence in Quantum Logic |
| title_full_unstemmed | State-Dependent Implication and Equivalence in Quantum Logic |
| title_short | State-Dependent Implication and Equivalence in Quantum Logic |
| title_sort | state dependent implication and equivalence in quantum logic |
| url | http://dx.doi.org/10.1155/2012/385341 |
| work_keys_str_mv | AT fedorherbut statedependentimplicationandequivalenceinquantumlogic |