Holographic entanglement entropy for measures other than the Von Neumann entropy

Question

In Ads-CFT, the Ryu-Takayanagi Entanglement entropy formula gives a nice geometric interpretation (in the bulk) for the entanglement of a region in a CFT. Also, it is much easier to calculate the entanglement using the same instead of other methods like the replica trick.

But, Von-Neumann entropy is one in several measures of entanglement. So, is it possible that other entanglement measures might have similar geometric interpretation in the bulk? One can axiomatically right down the properties that a "good" entanglement measure should satisfy. If would be interesting to know what those constraints correspond to in the bulk.

Answer 1

Let $\,\rho^{tot}\,$ be a pure state. Then every entanglement measure of its either subsystems due to the "Uniqueness Theorem for Entanglement Measures" coincide with the von Neumann entropy of its reduced density matrix [1]. This measure has a holographic dual via the R-T prescription. Now for $\,\rho^{tot}\,$ being a mixed state, one needs to use other measures than the von Neumann entropy like "Negativity", "Entanglement of Purification", "Relative Entropy", etc, and some of which are connected to each other with certain bounds and inequalities. The only measure for the mixed states which has been proposed (recently) to have a holographic dual is the Entanglement of Purification whose dual is the "Minimal Entanglement Wedge Cross-Section" [2] [3]. In [1], you can also find the necessary properties for a "good entanglement measure". I hope this answer helps you.

[1] https://arxiv.org/abs/quant-ph/0105017

[2] https://arxiv.org/abs/1708.09393

[3] https://arxiv.org/abs/1709.07424