Degradation of a quantum directional reference frame

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In this study, we are interested in the practical question of how many times a quantum directional reference frame (i.e., a spin-J system) can be used to perform a certain task with a given probability of success, under the assumption that the quantum directional reference frame evolves under a map that is covariant under rotations in SU(2). Our main theorem restricts the form of the state of the quantum reference frame as a function of how many times the covariant map was applied to it. Our results are a generalization of the paper of Bartlett el al. on the degradation of reference frames, and can be used to analyze certain types of interactions on a spin-J system.