Experimental tests of distributivity (Q1196081)

It is shown that the Tsirel'son inequality provides a test of the distributivity of the observable algebra, and thus applies also to Jordan algebras and distributive Segal algebras. Next, arbitrary empirical data on compatible observables are extended to a signed joint distribution for all observables. This signed distribution defines a state on an appropriate non-distributive algebra. In particular, there is a state on a non-distributive algebra which maximally violates the Tsirel'sons bound. Finally, it is also shown that Sherman's non-distributive Segal algebras are unsuitable as algebraic models of physical systems, since they do not possess enough compatible observables.