Extremal Covariant Positive Operator Valued Measures: the Case of a Compact Symmetry Group

In the modern theory of quantum mechanics, observables are represented as normalized positive operator valued measures (POVMs). The set of all POVMs having the same outcome space has natural convex structure. A convex mixture of two POVMs corresponds to a random choice between two measurement apparatuses. An extremal POVM thus describes an observable which is unaffected by this kind of randomness. In many applications one is interested in observables having some symmetry property. This is conventionally formulated as a covariance requirement with respect to a symmetry group. The covariance can arise from the symmetry of a particular problem [1] or the covariance can also be the defining property of some class of observables [2]. In this kind of situations the relevant set of observables is therefore the set of all covariant observables. The determination of extremal covariant POVMs has long been a problem [3]. In the case of a compact symmetry group G the following results have been obtained earlier: When G = T acts on itself and the representation of G has no multiplicities, the characterization of extremal covariant operator valued measures follows from [4, Theorem 1] if the representation space is finite-dimensional; in the infinite-dimensional case, the characterization is given in [5, Theorem 1]. If a compact group G has a finite-dimensional representation, the determination of extremals is solved in [6, Theorem 2] in the case of G acting on itself and in [7, Theorem 3] in the case of a transitive G-space. In this work we give a complete characterization of extremal covariant POVMs in the case of a compact group G and an arbitrary representation (Theorem 2 and Corollary 4). Our analysis of this problem proceeds in the following way. In Section 2 we fix the notations and recall some concepts which are essential for our investigation. In Section 3 we give two characterizations of the structure of covariant POVMs. We remark that the characterization by means of families of sesquilinear forms (Section 3.3) was already estabilished by Holevo in [8]. Finally, Section 4 contains the main results which characterizes the extremal covariant POVMs. These are obtained by setting up a correspondence between the set of covariant POVMs and