  We establish bounds on quantum correlations in many-body systems. They reveal
what sort of information about a quantum system can be simultaneously recorded
in different parts of its environment. Specifically, independent agents who
monitor environment fragments can eavesdrop only on amplified and redundantly
disseminated - hence, effectively classical - information about the
decoherence-resistant pointer observable. We also show that the emergence of
classical objectivity is signaled by a distinctive scaling of the conditional
mutual information, bypassing hard numerical optimizations. Our results
validate the core idea of Quantum Darwinism: objective classical reality does
not need to be postulated and is not accidental, but rather a compelling
emergent feature of quantum theory that otherwise - in absence of decoherence
and amplification - leads to "quantum weirdness". In particular, a lack of
consensus between agents that access environment fragments is bounded by the
information deficit, a measure of the incompleteness of the information about
the system.
