Universal behavior beyond multifractality of wave-functions at measurement--induced phase transitions

We numerically investigate the structure of many-body wave functions of 1D random quantum circuits with local measurements employing the participation entropies. The leading term in system size dependence of participation entropies indicates a multifractal scaling of the wave-functions at any non-zero measurement rate. The sub-leading term contains universal information about measurement--induced phase transitions and plays the role of an order parameter, being non-zero in the error-correcting phase and vanishing in the quantum Zeno phase. We provide an analytical interpretation of this behavior expressing the participation entropy in terms of partition functions of classical statistical models in 2D.