Relative Entanglement Entropies in 1+1-dimensional conformal field theories

TL;DR

This paper calculates the relative entanglement entropy between excited states in 1+1D conformal field theories using the replica trick, providing explicit results for free bosonic fields and validating them with numerical simulations.

Contribution

It introduces a method to compute relative entanglement entropies between excited states in 1+1D CFTs using the replica trick and applies it to free bosonic fields, with exact numerical validation.

Findings

Derived explicit formulas for relative entanglement entropy of excited states.

Validated theoretical predictions with numerical results in XX spin-chain.

Provided new insights into entanglement structure of primary operator states.

Abstract

We study the relative entanglement entropies of one interval between excited states of a 1+1 dimensional conformal field theory (CFT). To compute the relative entropy $S(\rho_1 \| \rho_0)$ between two given reduced density matrices $\rho_1$ and $\rho_0$ of a quantum field theory, we employ the replica trick which relies on the path integral representation of ${\rm Tr} ( \rho_1 \rho_0^{n-1} )$ and define a set of R\'enyi relative entropies $S_n(\rho_1 \| \rho_0)$. We compute these quantities for integer values of the parameter $n$ and derive via the replica limit, the relative entropy between excited states generated by primary fields of a free massless bosonic field. In particular, we provide the relative entanglement entropy of the state described by the primary operator $i \partial\phi$, both with respect to the ground state and to the state generated by chiral vertex operators. These predictions are tested against exact numerical calculations in the XX spin-chain finding perfect agreement.