标题： 测量诱导动力学与迹保留副本截断 显示英文标题

标题： Measurement Induced Dynamics and Trace Preserving Replica Cutoffs

摘要： 我们提出了一种通用的方法来解决在复制量子系统的测量诱导动力学中出现的无限层次问题。 我们的方法通过引入保迹的复制截断，并使用类似层析成像的技术，从较低阶的复制态估计较高阶的复制态，从而确保部分迹约简性质严格得到保持。 这保证了单复制系统动力学正确地还原为标准Lindblad演化。 通过系统地在不同阶的复制空间之间映射信息，我们刻画了部分迹运算下的零空间，并概述了有效算法的方法以强制实现正定性。 重要的是，研究表明，预先计算的自由费米子态的随机高斯系综提供了一种有效且计算高效的手段来稳定复制层次结构，即使在存在相互作用的情况下也是如此。 对小规模相互作用费米子系统的数值测试展示了我们方法的有效性和实用性，显示了与轨迹方法的高度一致，同时提供了显著更好的统计收敛性。 显示英文摘要

摘要： We present a general methodology for addressing the infinite hierarchy problem that arises in measurement-induced dynamics of replicated quantum systems. Our approach introduces trace-preserving replica cutoffs using tomographic-like techniques to estimate higher-order replica states from lower ones, ensuring that partial trace reduction properties are rigorously maintained. This guarantees that the dynamics of single-replica systems correctly reduce to standard Lindblad evolution. By systematically mapping information between replica spaces of different orders, we characterise null spaces under partial trace operations and outline efficient algorithmic approaches to enforce positivity. Importantly, it is demonstrated that pre-calculated stochastic Gaussian ensembles of free fermion states provide an effective and computationally efficient means to stabilise the replica hierarchy, even in the presence of interactions. Numerical tests on small interacting fermionic systems illustrate the effectiveness and practicality of our approach, showing precise agreement with trajectory methods while providing significantly better statistical convergence.