受监测自由费米子中的数波动与纠缠光谱参与度
Number Fluctuations and Entanglement-Spectrum Participation in Monitored Free Fermions
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中文总结 AI 辅助
研究显示，受监测自由费米子中，粒子数波动 F 和参与式有效纠缠模式数 M 在强监测下表现出统计关联，F 作为熵的自然伴侣，M 则为方差减少的数值总结。
中文摘要 AI 辅助
受监测的自由费米子链在测量率增加时表现出纠缠标度的转变，从弱监测的亚广延 regime 转变为强监测下的面积定律。我通过轨迹解析的关联矩阵模拟，对长度达到 L=96 的链进行测试，检验两个由单粒子纠缠光谱构建的诊断指标是否具有实际价值：粒子数波动 F_A=Σ_k ν_k(1-ν_k) 和参与式有效纠缠模式数 M_A=exp[-Σ_k w_k ln w_k]，其中 w_k ∝ ν_k(1-ν_k)。这两个量跟踪这种转变，随着测量概率单调递减，并在强监测 regime 中失去系统尺寸依赖性。然而，在轨迹和时间平均的稳态值层面，M 几乎是 F 的确定性函数：汇总曲线能捕捉到所有测量率下的 99.7% 方差，因此在该模型中 M 几乎不携带独立信息。其主要优势是统计性的：在强监测下，M 的轨迹到轨迹系数变异比熵或 F 的变异小约一个数量级。对子系统标度在 L=48-96 的模型比较进一步表明，在弱监测下，纯对数定律在这些尺寸上并不被统计偏好；仍存在一个小的准广延成分和漂移的有效对数系数。我得出结论：F 是熵的自然实验驱动的伴侣，而 M 最好被视为对相同信息的方差减少的数值总结，而非独立诊断。
英文摘要
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, $\mathcal{M}_A=\exp[-\sum_k w_k\ln w_k]$, with $w_k\proptoν_k(1-ν_k)$. 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.
