Chapter 20 - Measures on Quantum Structures

Abstract

This chapter describes the different aspects of measures on quantum structures. States on quantum structures are the basic way through which information can be derived from the physical system under measurement. States are roughly speaking probability measures and they describe the physical system. For classical Kolmogorov model, there are plenty of states that are σ-additive probability measures. The most important example of quantum logics is the quantum logic of a real, complex, or quaternion Hilbert space. The problem of existence of states on quantum logics, such as orthomodular lattices or orthomodular posets, showed that the situation with existence of states on quantum logic is a complicated problem, and it generated new ways of constructions of finite orthomodular lattices and posets, respectively. The linearity problem is important for noncommutative integration theory and noncommutative probability theory. The importance of MV-algebras for quantum structures follows from an important property that they model a classical situation with unsharp measurements. It is observed in the chapter that bounded commutative BCK-algebras are equivalent to MV-algebras.