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This paper presents a series of general results about the optimal estimationof physical transformations in a given symmetry group. In particular, it isshown how the different symmetries of the problem determine differentproperties of the optimal estimation strategy. The paper also contains adiscussion about the role of entanglement between the representation andmultiplicity spaces and about the optimality of square-root measurements.

Group theoretic structures in the estimation of an unknown unitary transformation