Unsharp number observable and Neumark theorem

摘要

An unsharp quantum observable represented by a commutativePOV measure F with real spectrum can be interpreted as the randomization of anobservable represented by a self-adjoint operator A, the sharp reconstruction of F.We consider a commutative POV measure {F_k} _(kEK),withKreal and countable,such that the operators Fk are discrete and prove that A can be interpreted as theprojection of a Neumark operator corresponding to the Neumark extension of F.The starting point is a relevant physical example, the unsharp number observable.Moreover, it is shown that the result is true for the unsharp position observable