Number Fluctuations and Entanglement-Spectrum Participation in Monitored Free Fermions

On finite system sizes, monitored one-dimensional free-fermion chains display a broad crossover in entanglement scaling as the measurement rate increases, from sub-extensive behavior at weak monitoring toward an area law at strong monitoring. Analytical field theory and recent large-scale simulations indicate that this apparent change is not a finite-rate transition in the thermodynamic limit. I test, using trajectory-resolved correlation-matrix simulations chains up to $L=96$ and up to 128 Born-rule trajectories per parameter point, whether two quantities built from the single-particle entanglement spectrum are useful finite-size diagnostics: the bipartite particle-number fluctuation F_A=sum_kν_k\(1-ν_k\) and a participation-style effective number of entangling modes, mathcal{M}_A=exp\[-sum_k w_kln w_k\], with w_kproptoν_k\(1-ν_k\). Both quantities track the crossover. However, for trajectory- and time-averaged steady-state values, mathcal{M} is nearly a deterministic function of F: a pooled curve explains 99.7% of its variance across all rates, so mathcal{M} carries little independent information. Its main advantage is statistical: its trajectory-to-trajectory coefficient of variation is up to a factor of sim2 smaller than that of the entropy or F under strong monitoring. Subsystem-scaling comparisons at L=48--96 also show that a pure logarithmic law is not statistically preferred under weak monitoring; a small residual quasi-extensive component and a drifting logarithmic coefficient are instead consistent with a crossover rather than a critical phase. Thus, F remains the natural experimentally motivated companion to the entropy, whereas mathcal{M} is best viewed as a variance-reduced numerical summary of essentially the same information.