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The dynamics of quantum information in many-body systems with large onsite Hilbert space dimension admits an enlightening description in terms of effective statistical mechanics models. Motivated by this fact, we reveal a connection between three separate models: the classically chaotic $d$-adic R\'{e}nyi map with stochastic control, a quantum analog of this map for qudits, and a Potts model on a random graph. The classical model and its quantum analog share a transition between chaotic and controlled phases, driven by a randomly applied control map that attempts to order the system. In the quantum model, the control map necessitates measurements that concurrently drive a phase transition in the entanglement content of the late-time steady state. To explore the interplay of the control and entanglement transitions, we derive an effective Potts model from the quantum model and use it to probe information-theoretic quantities that witness both transitions. The entanglement transition is found to be in the bond-percolation universality class, consistent with other measurement-induced phase transitions, while the control transition is governed by a classical random walk. These two phase transitions merge as a function of model parameters, consistent with behavior observed in previous small-size numerical studies of the quantum model., Comment: 14+3 pages, 13+1 figures, 3+4 tables

We study measurement-induced entanglement and control phase transitions in a quantum analog of the Bernoulli map subjected to a classically-inspired control protocol. When entangling gates are restricted to the Clifford group, separate entanglement ($p_\mathrm{ent}$) and control ($p_\mathrm{ctrl}$) transitions emerge, revealing two distinct universality classes. The control transition has critical exponents $\nu$ and $z$ consistent with the classical map (a random walk) while the entanglement transition is revealed to have similar exponents as the measurement-induced phase transition in Clifford hybrid dynamics. This is distinct from the case of generic entangling gates in the same model, where $p_\mathrm{ent} = p_\mathrm{ctrl}$ and universality is controlled by the random walk., Comment: 6 + 1 pages, 4 + 1 figures