Towards an entanglement measure for mixed states in CFTs based on relative entropy

Tadashi Takayanagi, Tomonori Ugajin, Koji Umemoto

Abstract Relative entropy of entanglement (REE) is an entanglement measure of bipartite mixed states, defined by the minimum of the relative entropy S(ρ AB ||σ AB ) between a given mixed state ρ AB and an arbitrary separable state σ AB . The REE is always bounded by the mutual information I AB = S(ρ AB ||ρ A ⊗ ρ B ) because the latter measures not only quantum entanglement but also classical correlations. In this paper we address the question of to what extent REE can be small compared to the mutual information in conformal field theories (CFTs). For this purpose, we perturbatively compute the relative entropy between the vacuum reduced density matrix ρ AB 0 on disjoint subsystems A ∪ B and arbitrarily separable state σ AB in the limit where two subsystems A and B are well separated, then minimize the relative entropy with respect to the separable states. We argue that the result highly depends on the spectrum of CFT on the subsystems. When we have a few low energy spectrum of operators as in the case where the subsystems consist of finite number of spins in spin chain models, the REE is considerably smaller than the mutual information. However in general our perturbative scheme breaks down, and the REE can be as large as the mutual information.

Mukund Rangamani, Tadashi Takayanagi

Yuya Kusuki, Kotaro Tamaoka

We derive dynamics of the entanglement wedge cross section directly from the two-dimensional holographic CFTs with a local operator quench. This derivation is based on the reflected entropy, a correlation measure for mixed states. We further compare these results with the mutual information and ones for integrable systems. This comparison directly suggests the classical correlation plays an important role in chaotic systems, unlike integrable ones. Besides a local operator quench, we study the reflected entropy in heavy primary states and find a breaking of the subsystem ETH. We checked the above results also hold for the odd entanglement entropy, which is another measure for mixed states related to the entanglement wedge cross section.

Benjamin R. Safdi

Aitor Lewkowycz, Robert Cobb Myers, Michael Smolkin

Debajyoti Sarkar, Souvik Banerjee, Johanna Erdmenger B

In the context of relating AdS/CFT to quantum information theory, we propose a holographic dual of Fisher information metric for mixed states in the boundary field theory. This amounts to a holographic measure for the distance between two mixed quantum states. For a spherical subregion in the boundary we show that this is related to a particularly regularized volume enclosed by the Ryu-Takayanagi surface. We further argue that the quantum correction to the proposed Fisher information metric is related to the quantum correction to the boundary entanglement entropy. We discuss consequences of this connection.