The connection between holographic entanglement and complexity of purification

TL;DR

This work ties together entanglement of purification and a holographic notion of purification complexity under the $E_P=E_W$ conjecture, using a two-strip boundary setup to map how EoP, CoP, and VI respond to dimensions, temperature, and geometry. It introduces two new mixed-state measures, CoP and VI, grounded in the CV proposal and interval-volume constructions, and analyzes their behavior across Schwarzschild-AdS, BTZ massive, charged BTZ, and multipartite geometries. The results show mass and charge generally suppress both EoP and CoP, with distinct signatures in VI that reveal quantum-locked correlations and nontrivial dependence on dimension and separation. The paper also provides LOCC and bit-thread interpretations, offering operational insight into how these correlation and complexity measures reflect underlying quantum information processing in holographic setups, and outlines rich avenues for future dynamical and background-generalizations.

Abstract

In this work we study how entanglement of purification (EoP) and the new quantity of "complexity of purification" are related to each other using the $E_P=E_W$ conjecture. First, we consider two strips in the same side of a boundary and study the relationships between the entanglement of purification of this mixed state and the parameters of the system such as dimension, temperature, length of the strips and the distance between them. Next, using the same setup, we introduce two definitions for the complexity of mixed states, complexity of purification (CoP) and the interval volume (VI). We study their connections to other parameters similar to the EoP case. Then, we extend our study to more general examples of BTZ black holes solution in massive gravity, charged black holes and multipartite systems. Finally, we give various interpretations of our results using resource theories such as LOCC and also bit thread picture.

Figures (39)

• Figure 1: Two strips of $A$ and $B$ with length $l$ and with the distance $D$ between them. The two turning points corresponding to region $ad$ and $bc$ are $m$ and $m'$ and $\Gamma$ is the cross section of "connected" entanglement wedge.
• Figure 2: The relationship between turning point and width of "one" strip.
• Figure 3: The relationship between entanglement entropy and the width of one strip $w$ and turning point $z_0$ for $d=3$.
• Figure 4: For each dimension, the region below the lines have non-vanishing EoP.
• Figure 5: The relationship between $D_c(d,\infty)$ and dimension $d$.