Number Fluctuations and Entanglement-Spectrum Participation in Monitored Free Fermions

Abstract

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 $F_A=\sum_k\nu_k(1-\nu_k)$ and a participation-style effective number of entangling modes, $\mathcal{M}_A=\exp[-\sum_k w_k\ln w_k]$, with $w_k\propto\nu_k(1-\nu_k)$. 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.