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Convertibility of Observables

Some problems of quantum information, cloning, estimation and testing of states, universal coding etc., are special example of the following `state convertibility' problem. In this paper, we consider the dual of this problem, 'observable conversion problem'. Given families of operators $\{L_θ}\}_{θ\inΘ}$ and $\{M_θ}\}_{θ\inΘ}$ , we ask whether there is a completely positive (sub) unital map which sends $\{L_θ}\}$ to $\{M_θ}\}$ for each θ. We give necessary and sufficient conditions for the convertibility in some special cases.