Entanglement Entropy of Eigenstates of Quantum Chaotic Hamiltonians

TL;DR

This paper investigates how the entanglement entropy of eigenstates in quantum chaotic systems approaches maximality, revealing a deviation that grows with system size and providing numerical evidence of bound saturation.

Contribution

It establishes an upper bound for the average entanglement entropy of particle-number conserving eigenstates, showing deviation growth with system volume, and confirms bound saturation through numerical analysis.

Findings

Deviation from maximal entropy grows with the square root of system volume.

Numerical results indicate the bound is saturated as system size increases.

Entanglement entropy approaches maximality in large quantum chaotic systems.

Abstract

In quantum statistical mechanics, it is of fundamental interest to understand how close the bipartite entanglement entropy of eigenstates of quantum chaotic Hamiltonians is to maximal. For random pure states in the Hilbert space, the average entanglement entropy is known to be nearly maximal, with a deviation that is, at most, a constant. Here we prove that, in a system that is away from half filling and divided in two equal halves, an upper bound for the average entanglement entropy of random pure states with a fixed particle number and normally distributed real coefficients exhibits a deviation from the maximal value that grows with the square root of the volume of the system. Exact numerical results for highly excited eigenstates of a particle number conserving quantum chaotic model indicate that the bound is saturated with increasing system size.