Abstract
The entanglement entropy of subsystems of typical eigenstates of quantum many-body Hamiltonians has recently been 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.
- Received 17 December 2021
- Revised 4 May 2022
DOI:https://doi.org/10.1103/PRXQuantum.3.030201
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
The entanglement entropy quantifies how measurements of two parts of a quantum system are correlated in a way that cannot be explained by classical physics. It has been extensively studied in ground states of physical systems, where it usually scales with the area of the boundary between the two parts of the system and has helped unveil unique properties of quantum phases of matter. The goal of this tutorial is to study the entanglement entropy of typical pure quantum states, which has been found to be the same as in highly-excited energy eigenstates of physical systems. In this regime, the entanglement entropy scales with the volume of the smaller of the two parts of the system and reveals valuable information about the dynamical properties of the physical system, e.g., if it behaves chaotically or not. We introduce tools from quantum information and random matrix theory that can be broadly used to analytically study the entanglement entropy of fermionic systems in arbitrary dimensions. We provide a pedagogical introduction to analytical calculations of the average entanglement entropy in six different ensembles of pure quantum states. The ensembles are either for general pure quantum states or for the subset of pure Gaussian states, and are further distinguished by whether the pure states have or do not have a fixed number of particles. The analytical results derived are shown to describe numerical results obtained for the entanglement entropy of highly-excited energy eigenstates of quantum-chaotic and integrable many-body quantum systems. Looking forward, the results and tools introduced in this tutorial are a step forward in establishing the entanglement entropy as a promising diagnostic of quantum chaos and integrability in many-body quantum systems.
