Clean Positive Operator Valued Measures

Abstract

In quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure (POVM). A quantum channel transforms POVM&#39;s into POVM&#39;s, generally irreversibly, thus loosing some of the information retrieved from the measurement. This poses the problem of which POVM&#39;s are &#34;undisturbed&#34;, namely they are not irreversibly connected to another POVM. We will call such POVM clean. In a sense, the clean POVM&#39;s would be &#34;perfect&#34;, since they would not have any additional &#34;extrinsical&#34; noise. Quite unexpectedly, it turns out that such cleanness property is largely unrelated to the convex structure of POVM&#39;s, and there are clean POVM&#39;s that are not extremal and vice-versa. In this paper we solve the cleannes classification problem for number n of outcomes n<=d (d dimension of the Hilbert space), and we provide a a set of either necessary or sufficient conditions for n>d, along with an iff condition for the case of informationally complete POVM&#39;s for n=d^2.