2026/07/08 by Enso O. Torres Alegre
#quant-ph
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 FA=∑kνk(1-νk) and a participation-style effective number of entangling modes, MA=exp[-∑k wkln wk], with wk∝νk(1-νk). Both quantities track the crossover. However, for trajectory- and time-averaged steady-state values, M is nearly a deterministic function of F: a pooled curve explains 99.7% of its variance across all rates, so M carries little independent information. Its main advantage is statistical: its trajectory-to-trajectory coefficient of variation is up to a factor of ∼2 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 M is best viewed as a variance-reduced numerical summary of essentially the same information.