State-Dependent Implication and Equivalence in Quantum Logic

Abstract EN

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.