Abstract According to the concept of typicality, an ensemble average can be accurately approximated by an expectation value with respect to a single pure state drawn at random from a high-dimensional Hilbert space. This random-vector approximation, or trace estimator, provides a powerful approach to, e.g. thermodynamic quantities for systems with large Hilbert-space sizes, which usually cannot be treated exactly, analytically or numerically. Here, we discuss the finite-size scaling of the accuracy of such trace estimators from two perspectives. First, we study the full probability distribution of random-vector expectation values and, second, the full temperature dependence of the standard deviation. With the help of numerical examples, we find pronounced Gaussian probability distributions and the expected decrease of the standard deviation with system size, at least above certain system-specific temperatures. Below and in particular for temperatures smaller than the excitation gap, simple rules are not available.

中文翻译：

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基于典型性的估计的有限尺度缩放

摘要 根据典型性的概念，对于从高维希尔伯特空间中随机抽取的单个纯态，可以通过期望值精确近似集合平均值。这种随机向量近似或迹估计器提供了一种强大的方法，例如用于具有大希尔伯特空间大小的系统的热力学量，这些系统通常不能被精确地、解析地或数值地处理。在这里，我们从两个角度讨论此类迹估计器精度的有限大小缩放。首先，我们研究了随机向量期望值的完整概率分布，其次，研究了标准偏差的完整温度依赖性。借助数值例子，我们发现明显的高斯概率分布和标准偏差随系统规模的预期下降，至少高于某些特定系统的温度。在低于激发间隙的温度下，尤其是对于小于激发间隙的温度，没有简单的规则可用。