Connection between joint distribution and compatibility
The paper dicusses the connections between the concept of joint distribution of observables on a quantum logic and that of compatibility of observables. Within the framework of axiomatic quantum theory a necessary and sufficient condition for the existence of the joint distribution of the observables \(x_ 1,...,x_ n\) is given. Namely, \(x_ 1,...,x_ n\) possess a joint distribution in a state m if and only if \[ \sum^{1}_{j_ 1,...,j_ n=0}m(\bigwedge^{n}_{i=1}x_ i(E_ i^{j_ i}))=1 \] for any Borel subsets \(E_ 1,...,E_ n\) of the real line \({\mathbb{R}}\) (by definition, \(E^ j=E\) if \(j=1\), and \(E^ j={\mathbb{R}}\setminus E\) if \(j=0)\). Lateron, a theorem is proved giving a necessary and sufficient condition for the existence of the joint distribution of the observables \(x_ 1,...,x_ n\) with pure point spectra. It is shown that the existence of the joint distribution in a state m implies that the corresponding measurements can be made simultaneously, due to the compatibility of the ''relativized observables''. In the final part of the paper three categories of compatibility of observables are defined: compatibility, partial compatibility, and total incompatibility. The paper ends with some interesting theorems on total incompatibility.
scientific article; zbMATH DE number 3878925