Quantum Darwinism and the quality of Petz recovery

According to Quantum Darwinism, system-environment interactions both einselect particular system properties and encode them redundantly in many independent subsets of the environment, called fragments. This redundancy implies that an observer can recover the einselected information by accessing just one such fragment. However, the protocol by which such reconstruction should occur is often left unspecified. Considering a system Γ interacting with a multipartite environment Ξ, we investigate whether, and under what conditions, the einselected state of Γ can be recovered from environmental fragments using the Petz recovery map. We show that the fidelity between the system's initial state and the state reconstructed via Petz recovery develops a plateau as a function of the fragment size. Our results are supported by both analytical arguments and numerical simulations of large but tractable models.