  We investigate the structure of many-body wave functions of 1D quantum
circuits with local measurements employing the participation entropies. The
leading term in system size dependence of participation entropy indicates a
model dependent 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 constant non-zero in the error correcting phase and vanishing in the
quantum Zeno phase. We provide robust numerical evidence investigating a
variety of quantum many-body systems, and provide an analytical interpretation
of this behavior expressing the participation entropy in terms of partition
functions of classical statistical models in 2D.
