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

TL;DR

This paper analyzes entanglement and purification-based quantities in two-interval vacuum states of 1+1D free QFTs (Klein–Gordon and Ising CFTs) using the most general Gaussian purifications. It develops a covariance-matrix framework to efficiently compute entanglement of purification and complexity of purification, benchmarking mutual information against CFT predictions and exploring large-distance scalings. The authors present detailed Gaussian-purification results for EoP and CoP in single and adjacent two-interval setups, including explicit fittings for bosons and fermions, and compare with holographic expectations and Fisher–Rao-type proposals. Subtle issues such as zero modes in the massless boson and locality differences under Jordan–Wigner mapping are carefully analyzed, offering a comprehensive picture of purification-based information measures in free QFTs and a scalable numerical approach for future studies of non-Gaussian and interacting cases.

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.

Figures (14)

• Figure 1: The subsystem that defines reduced density matrices for our discretized bosonic and fermionic models in their vacuum state consists of two intervals of a width of $w_{A}/\delta$ and $w_{B}/\delta$ sites and separated by a distance of $d/\delta$ sites, where $\delta$ is the lattice spacing. When $d = 0$, we will keep $w_{A}$ and $w_{B}$ generic. When $d > 0$, we will set for simplicity $w_{A} = w_{B} \equiv w$ and the natural continuum combination is $w/d$. We will see that numerically determined MI and EoP approach in the continuum limit functions of $w/d$. With CoP the situation is more complicated, as it turns out to be ultraviolet divergent and brings in an additional dimensionful scale through the class of reference states of interest $|\psi_R\rangle$.
• Figure 2: Numerical data for bosonic MI (a-d) and EoP (e-g) in the regimes of large (a,b,e) and small (c,d,f,g) ratio of block width $w$ to distance $d$ on a periodic system of $N$ sites of bosons. (a-b) Logarithmic coefficient of $S_{A\cup B}$ and $I(A:B)$. (c-d) Decay power and logarithmic coefficient of $I(A:B)$. (e) Logarithmic coefficient of $E_P$. (f-g) Decay power and logarithmic coefficient of $E_P$. Expected limits and numerical estimates are in table \ref{['tab:EoP-MI']}. The bosonic scale $m L = m N \delta$ is set to $10^{-5}$ for MI and $10^{-3}$ for EoP, as the MI computation is more stable at small values of $m L$.
• Figure 3: Bosonic ($c=1$, (a)) and fermionic/Ising spin EoP ($c=\tfrac{1}{2}$, (b)) for two adjacent ($d = 0$) subsystems $A$ and $B$ on $\frac{w_A+w_B}{\delta}=12$ sites, with the continuum result \ref{['EQ_EOP_A1A2t']} for a fitted lattice spacing $\epsilon$ plotted as a dashed curve. Total system size $N=1200$. Bosonic mass scale $m \, L = 10^{-4}$.
• Figure 4: Discrete derivative of fermionic (a) and bosonic squared CoP (b, c) of a single interval of length $\ell/\delta$. $m$ and $\delta$ are given in units where $\mu=1$. In both cases, the number of total sites is given by $N = 100\, w / \delta$.
• Figure 5: Divergences of bosonic CoP of a single interval of size $w$ at fixed $\mu\, \delta = 10^{-3}$ and $\frac{m}{\mu} \to 0$ (a) and at fixed $\frac{m}{\mu} = 10^{-3}$ and $\mu\, \delta \to 0$ (b), in units of $\mu=1$. In both cases, the leading divergence is linear. While the $\mu\, \delta$ divergence is $w$-dependent, the $\frac{m}{\mu}$ one is not.