Number Fluctuations and Entanglement-Spectrum Participation in Monitored Free Fermions

Abstract:Monitored free-fermion chains show a crossover in entanglement scaling as the measurement rate is increased, from a subextensive weak-monitoring regime to an area law at strong monitoring. I test, using trajectory-resolved correlation-matrix simulations of chains up to $L=96$, whether two diagnostics built from the single-particle entanglement spectrum add practical value: 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, decreasing monotonically with measurement probability and losing system-size dependence in the strongly monitored regime. However, at the level of trajectory- and time-averaged steady-state values, $\mathcal{M}$ is almost a deterministic function of $F$: a pooled curve captures 99.7\% of its variance across all measurement rates, so $\mathcal{M}$ carries little independent information in this model. Its main advantage is statistical: the trajectory-to-trajectory coefficient of variation of $\mathcal{M}$ is up to a factor of about 2 smaller than that of the entropy or $F$ at strong monitoring. A model comparison of subsystem scaling at $L=48$--$96$ further shows that, at weak monitoring, a pure logarithmic law is not statistically preferred at these sizes; a small residual quasi-extensive component and a drifting effective logarithmic coefficient remain. I conclude that $F$ is the natural experimentally motivated companion to the entropy, while $\mathcal{M}$ is best viewed as a variance-reduced numerical summary of largely the same information, not an independent diagnostic.

arXiv:2607.16278 [quant-ph]

(or arXiv:2607.16278v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.16278 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.