Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis

TL;DR

This paper investigates how reduced density matrices in 2D large central charge CFTs differ between states using various measures, and explores implications for the eigenstate thermalization hypothesis (ETH) in this context.

Contribution

It provides a detailed short interval expansion of multiple dissimilarity measures in 2D CFTs and applies these to analyze ETH for excited, thermal, and generalized Gibbs states.

Findings

Dissimilarity measures vanish when comparing excited states and GGE thermal states.

ETH is approximately valid for small subsystems and violated for large ones in 2D CFTs.

Derived Fisher information metric from dissimilarity measures enhances understanding of state differences.

Abstract

We calculate various quantities that characterize the dissimilarity of reduced density matrices for a short interval of length $\ell$ in a two-dimensional (2D) large central charge conformal field theory (CFT). These quantities include the R\'enyi entropy, entanglement entropy, relative entropy, Jensen-Shannon divergence, as well as the Schatten 2-norm and 4-norm. We adopt the method of operator product expansion of twist operators, and calculate the short interval expansion of these quantities up to order of $\ell^9$ for the contributions from the vacuum conformal family. The formal forms of these dissimilarity measures and the derived Fisher information metric from contributions of general operators are also given. As an application of the results, we use these dissimilarity measures to compare the excited and thermal states, and examine the eigenstate thermalization hypothesis (ETH) by showing how they behave in high temperature limit. This would help to understand how ETH in 2D CFT can be defined more precisely. We discuss the possibility that all the dissimilarity measures considered here vanish when comparing the reduced density matrices of an excited state and a generalized Gibbs ensemble thermal state. We also discuss ETH for a microcanonical ensemble thermal state in a 2D large central charge CFT, and find that it is approximately satisfied for a small subsystem and violated for a large subsystem.