zbMATH — the first resource for mathematics

On compatibility in quantum logics. (English) Zbl 0539.03045
The notions of compatibility of a given subset of the quantum logic L (i.e., the orthomodular \(\sigma\)-orthoposet) are presented and relationships between them are shown. Many modifications of this notion have been known in the literature and the authors here compare them. (i) \(A\subset L\) is said to be compatible if \(a\leftrightarrow b\) for any a,\(b\in A\); (ii) L satisfies the condition (c) if, for any a,b,\(c\in L\) which are mutually compatible, we have \(a\leftrightarrow b\vee c;\) (iii) \(A\subset L\) is strongly compatible if, for any a,\(b\in A\), \(a\leftrightarrow b\) and \(a=a_ 1\oplus c\), \(b=b_ 1\oplus c\), we have \(a_ 1,b_ 1,c\in L_ 0(A)\), where \(L_ 0(A)\) is the minimal sublogic of L generated by A; (iv) \(A\subset L\) is full compatible if, for any finite collection \(a_ 1,...,a_ n\in A,\) there is a finite subset of pairwise orthogonal elements, \(\{e_ i\}\), such that any \(a_ i=\bigvee_{j}e_{i_ j}\) for some subcollection of \(\{e_ i\}\). These notions are applied to the problem of the existence of the minimal sub-\(\sigma\)-algebra of L generated by A, and to the existence of the joint distribution of a given system of observables in a state.
Reviewer: A.Dvurečenskij

03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
81P20 Stochastic mechanics (including stochastic electrodynamics)