Entanglement and Complexity of Purification in (1+1)-dimensional free Conformal Field Theories

TL;DR

This paper investigates the entanglement and complexity of purifications for mixed states in (1+1)-dimensional free conformal field theories, using the most general Gaussian purifications to reveal universal properties and subtleties.

Contribution

It introduces the analysis of the most general Gaussian purifications for two intervals in free CFTs, providing new insights and comprehensive comparisons with existing results.

Findings

Identification of universal properties of purifications

Analysis of the massless limit and mutual information behavior

Insights into Hilbert space structure via Jordan-Wigner mapping

Abstract

Finding pure states in an enlarged Hilbert space that encode the mixed state of a quantum field theory as a partial trace is necessarily a challenging task. Nevertheless, such purifications play the key role in characterizing quantum information-theoretic properties of mixed states via entanglement and complexity of purifications. In this article, we analyze these quantities for two intervals in the vacuum of free bosonic and Ising conformal field theories using, for the first time, the~most general Gaussian purifications. We provide a comprehensive comparison with existing results and identify universal properties. We further discuss important subtleties in our setup: the massless limit of the free bosonic theory and the corresponding behaviour of the mutual information, as well as the Hilbert space structure under the Jordan-Wigner mapping in the spin chain model of the Ising conformal field theory.