Clean positive operator valued measures

In quantum mechanics the statistics of the outcomes of a measuring apparatusis described by a positive operator valued measure (POVM). A quantum channeltransforms POVM's into POVM's, generally irreversibly, thus loosing some of theinformation retrieved from the measurement. This poses the problem of whichPOVM's are "undisturbed", namely they are not irreversibly connected to anotherPOVM. We will call such POVM clean. In a sense, the clean POVM's would be"perfect", since they would not have any additional "extrinsical" noise. Quiteunexpectedly, it turns out that such cleanness property is largely unrelated tothe convex structure of POVM's, and there are clean POVM's that are notextremal and vice-versa. In this paper we solve the cleannes classificationproblem for number n of outcomes n<=d (d dimension of the Hilbert space), andwe provide a a set of either necessary or sufficient conditions for n>d, alongwith an iff condition for the case of informationally complete POVM's forn=d^2.Comment: Minor changes. amsart 21 pages. Accepted for publication on J. Math. Phy