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_\theta}\}_{\theta\in\Theta} and \{M_\theta}\}_{\theta\in\Theta} , we ask whether there is a completely positive (sub) unital map which sends \{L_\theta}\} to \{M_\theta}\} for each {\theta}. We give necessary and sufficient conditions for the convertibility in some special cases.