We report universal statistical properties displayed by ensembles of pure states that naturally emerge in quantum many-body systems. Specifically, two classes of state ensembles are considered: those formed by (i) the temporal trajectory of a quantum state under unitary evolution or (ii) the quantum states of small subsystems obtained by partial, local projective measurements performed on their complements. These cases, respectively, exemplify the phenomena of “Hilbert-space ergodicity” and “deep thermalization.” In both cases, the resultant ensembles are defined by a simple principle: The distributions of pure states have maximum entropy, subject to constraints such as energy conservation, and effective constraints imposed by thermalization. We present and numerically verify quantifiable signatures of this principle by deriving explicit formulas for all statistical moments of the ensembles, proving the necessary and sufficient conditions for such universality under widely accepted assumptions, and describing their measurable consequences in experiments. We further discuss information-theoretic implications of the universality: Our ensembles have maximal information content while being maximally difficult to interrogate, establishing that generic quantum state ensembles that occur in nature hide (scramble) information as strongly as possible. Our results generalize the notions of Hilbert-space ergodicity to time-independent Hamiltonian dynamics and deep thermalization from infinite to finite effective temperature. Our work presents new perspectives to characterize and understand universal behaviors of quantum dynamics using statistical and information-theoretic tools. Published by the American Physical Society 2024

中文翻译：

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深度热化和希尔伯特空间遍历性中的最大熵原理


我们报告了量子多体系统中自然出现的纯态系综所显示的普遍统计特性。具体来说，考虑了两类状态集成：由 （i） 量子态在幺正演化下的时间轨迹形成的，或 （ii） 通过对其补码进行部分局部射影测量而获得的小子系统的量子态。这些情况分别体现了“希尔伯特空间遍历性”和“深度热化”现象。在这两种情况下，所得的系集成由一个简单的原理定义：纯态的分布具有最大熵，受能量守恒等约束和热化施加的有效约束。我们通过推导出集合的所有统计时刻的显式公式，证明在广泛接受的假设下这种普遍性的必要和充分条件，并在实验中描述它们的可测量结果，来提出并数值验证该原理的可量化特征。我们进一步讨论了普遍性的信息论含义：我们的集合具有最大的信息内容，同时最难询问，从而确定了自然界中出现的通用量子态集合尽可能强烈地隐藏（扰乱）信息。我们的结果将希尔伯特空间遍历性的概念推广到与时间无关的哈密顿动力学和从无限到有限有效温度的深度热化。我们的工作为使用统计和信息论工具描述和理解量子动力学的普遍行为提供了新的视角。 美国物理学会 2024 年出版