Separate measurement- and feedback-driven entanglement transitions in the stochastic control of chaos

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_ent) and control (p_ctrl) transitions emerge, revealing two distinct universality classes. The control transition has critical exponents ν 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_ent=p_ctrl and universality is controlled by the random walk.