Volume-Law Entanglement Entropy of Typical Pure Quantum States
Volume-Law Entanglement Entropy of Typical Pure Quantum States
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DOI：
10.1103/prxquantum.3.030201
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发表时间：
2021-12
期刊：
影响因子：
9.7
通讯作者：
Eugenio Bianchi;L. Hackl;M. Kieburg;M. Rigol;L. Vidmar
中科院分区：
文献类型：
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作者：
Eugenio Bianchi;L. Hackl;M. Kieburg;M. Rigol;L. Vidmar
量子多体哈密顿算符的典型本征态的子系统的纠缠熵最近被证明是量子混沌和可积性的一个诊断。在量子混沌系统中，它表现为典型的纯态，而在可积系统中，它表现为典型的纯高斯态。在本教程中，我们提供了一个教学介绍的典型的纯态和典型的纯高斯态的子系统的纠缠熵的已知结果。他们都表现出领先的长期规模与体积的子系统，当小于一半的系统的体积，但体积法的前因子是根本不同的。对于典型的纯态，它是常数（并且是最大值），对于典型的纯高斯态，它取决于子系统和整个系统的体积之比。由于粒子数守恒在许多物理哈密顿量中起着重要的作用，我们讨论了它对典型纯态和典型纯高斯态的影响。我们证明，虽然领先的体积法条款的行为不会发生质的变化，subleading条款的性质可以改变。特别是，可能出现的子引导校正取决于子系统的体积的平方根。我们揭示了这些修正的起源。最后，我们讨论了典型的纯态的纠缠熵和随机矩阵理论的背景下得到的分析结果，以及物理哈密顿量的数值结果之间的联系。
The entanglement entropy of subsystems of typical eigenstates of quantum many-body Hamiltonians has been recently conjectured to be a diagnostic of quantum chaos and integrability. In quantum chaotic systems it has been found to behave as in typical pure states, while in integrable systems it has been found to behave as in typical pure Gaussian states. In this tutorial, we provide a pedagogical introduction to known results about the entanglement entropy of subsystems of typical pure states and of typical pure Gaussian states. They both exhibit a leading term that scales with the volume of the subsystem, when smaller than one half of the volume of the system, but the prefactor of the volume law is fundamentally different. It is constant (and maximal) for typical pure states, and it depends on the ratio between the volume of the subsystem and of the entire system for typical pure Gaussian states. Since particle-number conservation plays an important role in many physical Hamiltonians, we discuss its effect on typical pure states and on typical pure Gaussian states. We prove that while the behavior of the leading volume-law terms does not change qualitatively, the nature of the subleading terms can change. In particular, subleading corrections can appear that depend on the square root of the volume of the subsystem. We unveil the origin of those corrections. Finally, we discuss the connection between the entanglement entropy of typical pure states and analytical results obtained in the context of random matrix theory, as well as numerical results obtained for physical Hamiltonians.