  The theory of Quantum Darwinism aims to explain how our objective classical
reality arises from the quantum world, by analysing the distribution of
information about a quantum system that is accessible to multiple observers,
who probe the system by intercepting fragments of its environment. Previous
work showed that, when the number of environmental fragments grows, the quantum
channels modelling the information flow from system to observers become
arbitrarily close - in terms of diamond norm distance - to
"measure-and-prepare" channels, ensuring objectivity of observables; the
convergence is formalised by an upper bound on the diamond norm distance, which
decreases with increasing number of fragments. Here, we derive tighter diamond
norm bounds on the emergence of objectivity of observables for quantum systems
of infinite dimension, providing an approach which can bridge between the
finite- and the infinite-dimensional cases. Furthermore, we probe the tightness
of our bounds by considering a specific model of a system-environment dynamics
given by a pure loss channel. Finally, we generalise to infinite dimensions a
result obtained by Brand\~{a}o et al. [Nat. Commun. 6, 7908 (2015)], which
provides an operational characterisation of quantum discord in terms of
one-sided redistribution of correlations to many parties. Our results provide a
unifying framework to benchmark quantitatively the rise of objectivity in the
quantum-to-classical transition.
