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Quantum Darwinism (QD) proposes that classical objectivity emerges from thebroadcast of information about a microscopic degree of freedom into multiplefractions of a many-body environment. Such a broadcast of information is insharp contrast with its scrambling under strong interaction. It was recentlyshown that quantum dynamics interpolating between broadcasting and scramblingmay display sharp phase transitions of information propagation, namedQD-encoding transitions. Here, we initiate their systematic study in generic,non-Clifford settings. First, in a general theoretical setup where theinformation propagation is modeled as an isometry, whose input qudit isentangled with a reference, we propose a probe of the transitions -- thedistribution of the density matrix of the reference after measuring anenvironment fraction. This probe measures the classical correlation between thefraction and the injected information. We then apply the framework to twosimilar models defined by a tensor network on an expanding tree, modeling anoisy apparatus that attempts to broadcast the $z$ component of a spin-half. Wederive an exact recursion relation of the density matrix distribution, which weanalyze analytically and numerically. As a result we find three phases: QD,intermediate and encoding, and two continuous transitions. Theencoding-intermediate transition describes the establishment of nonzerocorrelation between the reference and a small environment fraction, and can beprobed by a ``coarse-grained'' measure of the total spin-$z$ of the fraction,which becomes non-Gaussian and symmetry breaking in the intermediate space. TheQD-intermediate transition is about whether the correlation is perfect. It mustbe probed by fined-grained measures, and corresponds to a more subtle symmetrybreaking in the replica space.

Quantum Darwinism-encoding transitions on expanding trees