We explore statistical fluctuations over the ensemble of quantum trajectories in a model of two-dimensional free fermions subject to projective monitoring of local charge across the measurement-induced phase transition. Our observables are the particle-number covariance between spatially separated regions, $G_{AB}$, and the two-point density correlation function, $\mathcal{C}\left(r\right)$. Our results exhibit a remarkable analogy to Anderson localization, with $G_{AB}$ corresponding to two-terminal conductance and $\mathcal{C}\left(r\right)$ to two-point conductance, albeit with different replica limits and unconventional symmetry class, geometry, and boundary conditions. In the delocalized phase, $G_{AB}$ exhibits “universal,” nearly Gaussian, fluctuations with variance of order unity. In the localized phase, we find a broad distribution of $G_{AB}$ with $\overline{-ln{G}_{AB}}\sim L$ (where $L$ is the system size) and the variance $\mathrm{var}(ln{G}_{AB})\sim L^{\mu }$, and similarly for $\mathcal{C}\left(r\right)$, with $\mu \approx 0.5$. At the transition point, the distribution function of $G_{AB}$ becomes scale invariant and $\mathcal{C}\left(r\right)$ exhibits multifractal statistics, $\overline{{\mathcal{C}}^q\left(r\right)}\sim r^{-q(d+1)-{\mathrm{\Delta }}_q}$. We characterize the spectrum of multifractal dimensions ${\mathrm{\Delta }}_q$. Our findings lay the groundwork for mesoscopic theory of monitored systems, paving the way for various extensions.

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• Dynamical phase transitions
• Nonlinear sigma model
• Open quantum systems
• Phase transitions
• Quantum correlations in quantum information
• Quantum entanglement
• Quantum measurements
• Quantum stochastic processes
• Multifractal systems
• Nonequilibrium systems
• Exact diagonalization
• Free-electron model
• Path-integral methods
• Renormalization group
• Tight-binding model