Information transfer in generalized probabilistic theories based on weak repeatability

Information transfer in generalized probabilistic theories (GPT) is an important problem. We have dealt with the problem based on repeatability postulate, which generalizes Zurek's result to the GPT framework [Phys. Lett. A \textbf{379} (2015) 2694]. A natural question arises: can we deduce the information transfer result under weaker assumptions? In this paper, we generalize Zurek's result to the framework of GPT using weak repeatability postulate. We show that if distinguishable information can be transferred from a physical system to a series of apparatuses under the weak repeatability postulate in GPT, then the initial states of the physical system must be completely distinguishable. Moreover, after each step of invertible transformation, the composite states of the composite system composed of the physical systems and the apparatuses must also be completely distinguishable.