# Subsystem trace distance in low-lying states of $(1+1)$-dimensional conformal field theories

## TL;DR

This paper develops a systematic replica method to compute the subsystem trace distance in (1+1)-dimensional conformal field theories, providing exact formulas and numerical validation for low-lying states in critical quantum systems.

## Contribution

It completes and extends previous work by deriving detailed formulas for trace distances in CFTs, including exact results for special cases and general expansions for arbitrary interval lengths.

## Key findings

- Exact trace distance formulas for special cases in free boson and fermion theories.
- Numerical validation showing perfect agreement with analytical predictions.
- New results for relative entropy and fidelity in 2D CFTs.

## Abstract

We report on a systematic replica approach to calculate the subsystem trace distance for a quantum field theory. This method has been recently introduced in [J. Zhang, P. Ruggiero, P. Calabrese, Phys. Rev. Lett. 122, 141602 (2019)], of which this work is a completion. The trace distance between two reduced density matrices $\rho_A$ and $\sigma_A$ is obtained from the moments $\textrm{tr} (\rho_A-\sigma_A)^n$ and taking the limit $n\to1$ of the traces of the even powers. We focus here on the case of a subsystem consisting of a single interval of length $\ell$ embedded in the low lying eigenstates of a one-dimensional critical system of length $L$, a situation that can be studied exploiting the path integral form of the reduced density matrices of two-dimensional conformal field theories. The trace distance turns out to be a scale invariant universal function of $\ell/L$. Here we complete our previous work by providing detailed derivations of all results and further new formulas for the distances between several low-lying states in two-dimensional free massless compact boson and fermion theories. Remarkably, for one special case in the bosonic theory and for another in the fermionic one, we obtain the exact trace distance, as well as the Schatten $n$-distance, for an interval of arbitrary length, while in generic case we have a general form for the first term in the expansion in powers of $\ell/L$. The analytical predictions in conformal field theories are tested against exact numerical calculations in XX and Ising spin chains, finding perfect agreement. As a byproduct, new results in two-dimensional CFT are also obtained for other entanglement-related quantities, such as the relative entropy and the fidelity.