The connection between holographic entanglement and complexity of purification

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.

Complexity and emergence of warped AdS3 space-time from chiral Liouville action

“…In [63], the connection between entanglement of purification (EoP) and complexity of purification (CoP) has been studied. These studies could also be done for the case warped conformal field theories and check how the lack of dilation in one direction could change the EoP and CoP and their evolutions [62].…”

“…In the NMG action, the mass of graviton plays the essential role which affects various quantum information quantities of the model, such as entanglement and its evolutions during quench processes, entanglement and complexity of purification, etc, which were studied in more details in [24,37,62,63].…”

Holographic subregion complexity in general Vaidya geometry

“…Finally, it is interesting to investigate the dynamical behavior of some new quantities which are analogous to the notions in the context of holographic entanglement entropy, such as mutual complexity [7] and complexity purification [5,[28][29][30].…”

Evolutions of entanglement and complexity after a thermal quench in massive gravity theory

“…For instance, one of the most important quantum information quantity for mixed states is entanglement of purification (EoP) introduced in [4], where its holographic dual has been considered to be the minimal entanglement wedge cross section [5]. Additionally, bit thread formalism for studying EoP has been addressed in [6][7][8][9]. Another one is complexity of purification(CoP) introduced in [9,10], which is the minimum number of gates needed to purify a mixed state.…”

“…Additionally, bit thread formalism for studying EoP has been addressed in [6][7][8][9]. Another one is complexity of purification(CoP) introduced in [9,10], which is the minimum number of gates needed to purify a mixed state. Next but not the last interesting quantum information quantity is the logarithmic negativity, which is a quantum entanglement measure for mixed quantum states and only captures the "quantum correlations" with the nature of "entanglement" [11].…”