A unified framework for the algebra of unsharp quantum mechanics

On the basis of the concrete operations definable on the set of effect operators on a Hilbert space, an abstract algebraic structure of sum Brouwer-Zadeh (SBZ)-algebra is introduced. This structure consists of a partial sum operation and two mappings which turn out to be Kleene and Brouwer unusual orthocomplementations. The Foulis-Bennett effect algebra substructure induced by any SBZ-algebra, allows one to introduce the notions of unsharp “state” and “observable” in such a way that any “state-observable” composition is a standad probability measure (classical state). The Cattaneo-Nisticò BZ substructure induced by any SBZ-algebra permits one to distinguish, in an equational and simple way, the sharp elements from the really unsharp ones. The family of all sharp elements turns out to be a Foulis-Randall orthoalgebra. Any unsharp element can be “roughly” approximated by a pair of sharp elements representing the best sharp approximation from the bottom and from the top respectively, according to an abstract generalization introduced by Cattaneo of Pawlack “rough set” theory (a generalization of set theory, complementary to fuzzy set theory, which describes approximate knowledge with applications in computer sciences). In both the concrete examples of fuzzy sets and effect operators the “algebra” of rough elements shows a weak SBZ structure (weak effect algebra plus BZ standard poset) whose investigation is set as an interesting open problem.