Volume-law entanglement entropy of typical pure quantum states

TL;DR

This paper reviews the entanglement entropy behavior in typical pure quantum states and Gaussian states, highlighting differences in volume-law scaling, effects of particle-number conservation, and connections to random matrix theory.

Contribution

It provides a comprehensive analysis of volume-law entanglement entropy in typical pure states and Gaussian states, including subleading corrections and the impact of conservation laws.

Findings

Typical pure states exhibit maximal volume-law entanglement entropy.

Gaussian states show volume-law scaling with a ratio-dependent prefactor.

Subleading corrections can depend on the square root of the subsystem volume.

Abstract

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.