Relative entropy of excited states in two dimensional conformal field theories

TL;DR

This paper derives a universal formula for the relative entropy between excited states in 2D conformal field theories, relating it to trace distance and confirming results with specific models.

Contribution

It provides a general formula for the relative entropy between primary states with the same conformal dimension in 2D CFTs, including universal behavior and checks with known models.

Findings

Relative entropy is proportional to trace square distance for same conformal dimension states.

Derived a universal leading term for different conformal dimensions in small interval expansion.

Confirmed formulas with calculations in free fields and the critical Ising model.

Abstract

We study the relative entropy and the trace square distance, both of which measure the distance between reduced density matrices of two excited states in two dimensional conformal field theories. We find a general formula for the relative entropy between two primary states with the same conformal dimension in the limit of a single small interval and find that in this case the relative entropy is proportional to the trace square distance. We check our general formulae by calculating the relative entropy between two generalized free fields and the trace square distance between the spin and disorder operators of the critical Ising model. We also give the leading term of the relative entropy in the small interval expansion when the two operators have different conformal dimensions. This turns out to be universal when the CFT has no primaires lighter than the stress tensor. The result reproduces the previously known special cases.