  Quantum Darwinism posits that the emergence of a classical reality relies on
the spreading of classical information from a quantum system to many parts of
its environment. But what are the essential physical principles of quantum
theory that make this mechanism possible? We address this question by
formulating the simplest instance of Darwinism -- CNOT-like fan-out
interactions -- in a class of probabilistic theories that contain classical and
quantum theory as special cases. We determine necessary and sufficient
conditions for any theory to admit such interactions. We find that every
non-classical theory that admits this spreading of classical information must
have both entangled states and entangled measurements. Furthermore, we show
that Spekkens' toy theory admits this form of Darwinism, and so do all
probabilistic theories that satisfy principles like strong symmetry, or contain
a certain type of decoherence processes. Our result suggests the
counterintuitive general principle that in the presence of local
non-classicality, a classical world can only emerge if this non-classicality
can be "amplified" to a form of entanglement.
