Holographic Entanglement of Purification from Conformal Field Theories

Abstract

We explore a conformal field theoretic interpretation of the holographic entanglement of purification, which is defined as the minimal area of entanglement wedge cross section. We argue that in AdS3/CFT2, the holographic entanglement of purification agrees with the entanglement entropy for a purified state, obtained from a special Weyl transformation, called path-integral optimizations. By definition, this special purified state has the minimal path-integral complexity. We confirm this claim in several examples. arXiv:1812.05268v1 [hep-th]

Some aspects of entanglement wedge cross-section

“…Also using a generalization of the above proposal to time dependent backgrounds, [14,21] studied the evolution of EoP after a quantum quench in the dual field theory. Related investigations attempting to better understand EoP both in the field theory and holography have also appeared in [22][23][24][25][26][27][28][29][30][31][32].…”

On the role of counterterms in holographic complexity

“…The discovery of Ryu-Takayanagi [3,4,6] and HRT formulas [5,7] marked the beginning of a new era during which we have been encountering increasing evidence [3][4][5][6][7][8][9][10][13][14][15][16][17][18][19][20][21][22] that quantum gravity and quantum information theory are two indispensable topics. The connection is strong enough that one might introduce a dictionary between them and apply the AdS/CFT correspondence [23] to calculate diverse quantities such as entanglement entropy [3-5, 8, 9], mutual information [10][11][12] and entanglement of purification [13][14][15][16]. Another important concept in quantum information theory is computational complexity which is believed to be useful in understanding the interior of black holes [17][18][19][20]22].…”

Holographic complexity from Crofton’s formula in Lorentzian AdS3

“…It is not clear to us why the number density of geodesic actually accounts for the cost. The good news is that the former does follow from entanglement entropy associated with the quantum circuit, which was computed in [30]. The conformal factors e 2φ at the end points provide corrections c/6 φ to the entanglement entropy of a single interval…”