Defuzzification and Joint Measurability of Quantum Fuzzy Observables

Commutative positive operator-valued measures (POVMs) are fuzzifications of spectral measures. In quantum mechanics, this corresponds to a connection between commutative unsharp observables (represented by commutative POVMs) and sharp observables (represented by self-adjoint operators); the former being a fuzzification of the latter. We prove that commutative unsharp observables can be defuzzified in order to obtain the sharp observables of which they are the fuzzy versions. We prove this in the case of POVMs defined on a general topological space which we require to be second countable and metrizable, generalizing some previous results on real POVMs. Then, we analyze some of the consequences of this defuzzification procedure. In particular we show that the joint measurability of two commutative (but generally not commuting) POVMs F1 and F2 corresponds to the existence of two commuting self-adjoint operators A+ 1 and A+ 2 in an extended Hilbert space H+ whose projections are the sharp versions of F1 and F2, respectively. In other words, the joint measurability of F1 and F2 is translated in the commutativity of A+ 1 and A+ 2 . This is proved for POVMs on a second countable, Hausdorff, locally compact topological space, generalizing similar results obtained in the case of real POVMs.