Glimpse of objectivity in bipartite systems for nonentangling pure dephasing evolutions

We study separable system-environment evolutions of pure dephasing type in the context of objectivity. We find that it can lead to the natural emergence of spectrum broadcast structures (SBSs), but at discrete time instances. Contrary to the standard way of obtaining SBS states, which requires entanglement, reaching such states here does not require decoherence (no unobserved environments are necessary). Yet the biggest difference is the basis with respect to which the SBS states are formed. Here it is not the pointer basis of the system given by the interaction with the environment, but their equal superposition basis, the so-called mutually unbiased basis. The price to pay is a simple SBS structure with just one environment and its momentary character.