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 and stochastic analog of the Bernoulli map subjected to a classically inspired control protocol. When entangling gates are restricted to the Clifford group, separate entanglement (pent) and control (pctrl) 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 with nonlocal control in the same model, where pent=pctrl and universality is controlled by the random walk.