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Clean Positive Operator Valued Measures

F. Buscemi, G. M. D'Ariano, M. Keyl, P. Perinotti, R. Werner

quant-pharXiv:quant-ph/0505095

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's into POVM's, generally irreversibly, thus loosing some of the information retrieved from the measurement. This poses the problem of which POVM'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'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's, and there are clean POVM'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's for n=d2.

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