We study the heat statistics of a multilevel $N$-dimensional quantum system monitored by a sequence of projective measurements. The late-time, asymptotic properties of the heat characteristic function are analyzed in the thermodynamic limit of a high, ideally infinite, number $M$ of measurements $(M\to \infty )$. In this context, the conditions allowing for an infinite-temperature thermalization (ITT), induced by the repeated monitoring of the quantum system, are discussed. We show that ITT is identified by the fixed point of a symmetric random matrix that models the stochastic process originated by the sequence of measurements. Such fixed point is independent on the nonequilibrium evolution of the system and its initial state. Exceptions to ITT, which we refer to as partial thermalization, take place when the observable of the intermediate measurements is commuting (or quasicommuting) with the Hamiltonian of the quantum system or when the time interval between measurements is smaller or comparable with the system energy scale (quantum Zeno regime). Results on the limit of infinite-dimensional Hilbert spaces ($N\to \infty$), describing continuous systems with a discrete spectrum, are also presented. We show that the order of the limits $M\to \infty$ and $N\to \infty$ matters: When $N$ is fixed and $M$ diverges, then ITT occurs. In the opposite case, the system becomes classical, so that the measurements are no longer effective in changing the state of the system. A nontrivial result is obtained fixing $M/N^2$ where instead partial ITT occurs. Finally, an example of partial thermalization applicable to rotating two-dimensional gases is presented.

中文翻译：

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多级系统中由量子监测引起的热化过程

我们研究多层次的热统计 $N$由一系列投影测量监控的维量子系统。热特征函数的后期渐近性质在一个高的、理想的无穷大数的热力学极限下进行分析$米$ 测量值 $(米\to \infty )$. 在这种情况下，讨论了由重复监测量子系统引起的无限温度热化 (ITT) 的条件。我们表明，ITT 是由对称随机矩阵的不动点识别的，该矩阵对由测量序列产生的随机过程进行建模。这样的不动点与系统的非平衡演化及其初始状态无关。ITT 的例外，我们称之为部分热化，发生在中间测量的可观察量与量子系统的哈密顿量交换（或准交换）时，或者当测量之间的时间间隔较小或与系统能量尺度相当时（量子芝诺政权）。无限维希尔伯特空间极限的结果 ($N\to \infty$)，描述具有离散谱的连续系统，也被提出。我们证明极限的顺序$米\to \infty$ 和 $N\to \infty$ 事项：当 $N$ 是固定的并且 $米$发散，然后发生 ITT。在相反的情况下，系统变得经典，因此测量不再有效地改变系统的状态。获得了一个非平凡的结果$米/N^2$而是发生部分 ITT。最后，给出了一个适用于旋转二维气体的局部热化的例子。