Entanglement and objectivity in pure dephasing models

We study the relation between the emergence of objectivity and qubit-environment entanglement generation. We find that although entanglement with the unobserved environments is irrelevant (since sufficiently strong decoherence can occur regardless), entanglement with the observed environments is crucial. In fact, the appearance of an objective qubit-observed-environment state is strictly impossible if their joint evolution does not lead to entanglement. Furthermore, if a single observer has access to a single environment (no macrofractions), then the required orthogonality of the observed environmental states comes only as a consequence of the system-environment state becoming strongly entangled (maximally entangled for the given initial occupation of the qubit if the environmental state is initially pure).