  We investigate how much information about a quantum system can be
simultaneously communicated to independent observers, by establishing
quantitative limits to bipartite quantum correlations in many-body systems. As
recently reported in Phys. Rev. Lett. 129, 010401 (2022), bounds on quantum
discord and entanglement of formation between a single quantum system and its
environment, e.g., a large number of photons, dictate that independent
observers which monitor environment fragments inevitably acquire only classical
information about the system. Here, we corroborate and generalize those
findings. First, we calculate continuity bounds of quantum discord, which
establish how much states with a small amount of quantum correlations deviate
from being embeddings of classical probability distributions. Also, we
demonstrate a universally valid upper bound to the bipartite entanglement of
formation between an arbitrary pair of components of a many-body quantum
system. The results confirm that proliferation of classical information in the
Universe suppresses quantum correlations.
