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Advances in Mathematical Physics
Volume 2012 (2012), Article ID 385341, 23 pages
Research Article

State-Dependent Implication and Equivalence in Quantum Logic

Department of Mathematics, Physics, and Geological Sciences, Serbian Academy of Sciences and Arts, Knez Mihajlova 35, 11000 Belgrade, Serbia

Received 29 March 2012; Accepted 9 September 2012

Academic Editor: Emilio Elizalde

Copyright © 2012 Fedor Herbut. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.