Entanglement of Purification in Free Scalar Field Theories

TL;DR

This work computes the entanglement of purification $E_P(\rho_{AB})$ for the ground state of a 1+1D free scalar field by discretizing the theory on a lattice and assuming a minimal Gaussian purification. The authors develop a tractable framework using a minimal Gaussian ansatz to minimize $S_{A\tilde{A}}$ and obtain explicit numerical results across several subsystem sizes and masses, observing a holographic-like plateau at small separations and a decay with distance. They compare $E_P$ to mutual information, analyze mutual information in purified spaces, and study monogamy and strong superadditivity, finding violations across many masses but restoration in the heavy-mass limit. The results provide a first quantitative glimpse of EoP in quantum field theories, offering upper bounds and connections to holographic pictures and motivating future continuum analyses in conformal field theories.

Abstract

We compute the entanglement of purification (EoP) in a 2d free scalar field theory with various masses. This quantity measures correlations between two subsystems and is reduced to the entanglement entropy when the total system is pure. We obtain explicit numerical values by assuming minimal gaussian wave functionals for the purified states. We find that when the distance between the subsystems is large, the EoP behaves like the mutual information. However, when the distance is small, the EoP shows a characteristic behavior which qualitatively agrees with the conjectured holographic computation and which is different from that of the mutual information. We also study behaviors of mutual information in purified spaces and violations of monogamy/strong superadditivity.

Figures (17)

• Figure 1: Holographic entanglement of purification. The shaded region is the entanglement wedge of the subsystems $A$ and $B$ in holographic CFTs (we take a constant time slice of global AdS). The dotted lines are the minimal surface whose area gives $S_{AB}$. The entanglement wedge cross-section $E_{W}(A:B)$ is defined by the minimal area (divided by $4G_{N}$) of codimension-2 surfaces which divide the entanglement wedge into two parts. In this figure this minimal surface is denoted by $\Sigma_{AB}^{*}$ and $E_{W}(A:B)={{\rm Area}(\sum_{AB}^{*})\over 4G_{N}}$.
• Figure 2: The setup for the computation of the holographic EoP $E_{W}(A:B)$ in Poincaré AdS$_3$ (the left picture), and the plots of $E_{W}(A:B)$ (the blue curve in the right picture) and half of holographic mutual information $I(A:B)$ (the orange curve in the right picture) as the functions of the distance ($d$) between $A$ and $B$. Both holographic EoP and mutual information show phase transition behaviors, though only the EoP is discontinuous. We set ${c\over 6}=1$ and the size $l=1$ with the transition point $d_{*}=\sqrt{2}-1$. After the phase transition, EoP and mutual information become zero.
• Figure 3: A derivation of $E_{P}=E_{W}$ based on the tensor network description of AdS space. We regard $A\tilde{A}B\tilde{B}$ as a new boundary of bulk spacetime defining an extended field theory. The subsystems $\tilde{A}$ and $\tilde{B},$ lying on the minimal surface used for computing $S_{AB},$ are identified with the ancilla system. The dashed lines denotes the minimal surfaces whose areas give $S_A$ or $S_B$, respectively. Now we have to minimize the $S_{A\tilde{A}}$ and that is achieved by minimizing the cross-section of the wedge and that surface is denoted by the thick green line.
• Figure 4: The holographic mutual information $I(A:\tilde{A})$ subtracted by $2S_{A}$. We set ${c\over 6}=1$ and the size $l=1$. It monotonically increases if we do not care about the phase transition at $d_{*}=\sqrt{2}-1$.
• Figure 5: An example of the setup for our lattice model. We set $N=16$ and took $|A|=|B|=2$. The distance $d$ between $A$ and $B$ is $d=3$. The complement of $A$ and $B$, called $C$, consists of twelve lattice sites.