State‐Dependent Implication and Equivalence in Quantum Logic

Fedor Herbut

Advances in Mathematical Physics, 2012, vol. 2012, issue 1

Abstract: Ideal occurrence of an event (projector) E leads to the known change of a state (density operator) ρ into EρE/[tr(Eρ)] (the Lüders state). It is shown that two events E and F give the same Lüders state if and only if the equivalence relation Eρ = Fρ 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 Q- of the Lüders state can be evaluated as Q-=E-(E⋀Q0), where ⋀ denotes the greatest lower bound, and Q0 is the null projector of ρ. State‐dependent implication ≤ρ extends absolute implication (which, in turn, determines the entire structure of quantum logic). Q- and ≤ρ are investigated in a closely related way to mutual benefit. Inherent in the preorder ≤ρ is the state‐dependent equivalence ~ρ, 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.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlamp:v:2012:y:2012:i:1:n:385341

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