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The linear cross-entropy (LXE) has been recently proposed as a scalable probeof the measurement-driven phase transition between volume- andarea-law-entangled phases of pure-state trajectories in certain monitoredquantum circuits. Here, we demonstrate that the LXE can distinguish distinctarea-law-entangled phases of monitored circuits with symmetries, and extractuniversal behavior at the critical points separating these phases. We focus on(1+1)-dimensional monitored circuits with an on-site $\mathbb{Z}_{2}$ symmetry.For an appropriate choice of initial states, the LXE distinguishes thearea-law-entangled spin glass and paramagnetic phases of the monitoredtrajectories. At the critical point, described by two-dimensional percolation,the LXE exhibits universal behavior which depends sensitively on boundaryconditions, and the choice of initial states. With open boundary conditions, weshow that the LXE relates to crossing probabilities in critical percolation,and is thus given by a known universal function of the aspect ratio of thedynamics, which quantitatively agrees with numerical studies of the LXE atcriticality. The LXE probes correlations of other operators in percolation withperiodic boundary conditions. We show that the LXE is sensitive to the richerphase diagram of the circuit model in the presence of symmmetric unitary gates.Lastly, we consider the effect of noise during the circuit evolution, andpropose potential solutions to counter it.

Universality of the cross entropy in $\mathbb{Z}_2$ symmetric monitored quantum circuits