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arXiv:1809.09109 · Physical Review Letters

Entanglement Wedge Cross Section from the Dual Density Matrix

Kotaro Tamaoka

Published Apr 9, 2019• 157 citations
arXivHTMLPDF

Abstract

We define a new information theoretic quantity called odd entanglement entropy (OEE) which enables us to compute the entanglement wedge cross section in holographic conformal field theories (CFTs). The entanglement wedge cross section has been introduced as a minimal cross section of the entanglement wedge, a natural generalization of the Ryu-Takayanagi surface. By using the replica trick, we explicitly compute the OEE for two-dimensional holographic CFT (three-dimensional anti-de Sitter space and planar Bañados-Teitelboim-Zanelli black hole) and see agreement with the entanglement wedge cross section. We conjecture this relation will hold in general dimensions.

DOI
10.1103/physrevlett.122.141601

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We define a new information theoretic quantity called odd entanglement entropy (OEE) which enables us to compute the entanglement wedge cross section in holographic conformal field theories (CFTs). The entanglement wedge cross section has been introduced as a minimal cross section of the entanglement wedge, a natural generalization of the Ryu-Takayanagi surface. By using the replica trick, we explicitly compute the OEE for two-dimensional holographic CFT (three-dimensional anti-de Sitter space and planar Bañados-Teitelboim-Zanelli black hole) and see agreement with the entanglement wedge cross section. We conjecture this relation will hold in general dimensions.
Abstract We obtain the odd entanglement entropy (OEE) for bipartite mixed states in ( 1 + 1 ) -dimensional boundary conformal field theories (BCFT 2 s) utilizing an appropriate replica technique. For the generic configuration of two disjoint intervals, the large- c factorization of the twist field correlator leads to several distinct phases depending on the relative sizes and separations of the intervals. The resulting expressions match consistently with the entanglement wedge cross section (EWCS) for the bulk AdS 3 braneworld geometries, thereby extending the proposed holographic duality between the difference of the OEE and the entanglement entropy with the EWCS to BCFT setups. Furthermore, we obtain the OEE for these mixed states in related holographic moving mirror models which simulate Hawking radiation from black holes and verify the above duality in these time-dependent backgrounds. In this context we analyze the time evolution of the OEE and establish the analogues of Page curves for this mixed-state correlation measure.
We construct a replica technique to perturbatively compute the odd entanglement entropy (OEE) for bipartite mixed states in TT‾\mathrm{T}\overline{\mathrm{T}}TT deformed CFT2s{\mathrm{CFT}}_{2}\mathrm{s}CFT2​s. This framework is then utilized to obtain the leading-order correction to the OEE for two disjoint intervals, two adjacent intervals, and a single interval in TT‾\mathrm{T}\overline{\mathrm{T}}TT deformed thermal CFT2s{\mathrm{CFT}}_{2}\mathrm{s}CFT2​s in the large central charge limit. The field theory results are subsequently reproduced in the high-temperature limit from holographic computations for the entanglement wedge cross sections in the dual bulk finite cutoff Ba\~nados-Teitelboim-Zanelli geometries. We further show that for finite size TT‾\mathrm{T}\overline{\mathrm{T}}TT deformed CFT2s{\mathrm{CFT}}_{2}\mathrm{s}CFT2​s at zero temperature the corrections to the OEE are vanishing to the leading order from both field theory and bulk holographic computations.
A recent proposal by Ryu and Takayanagi for a holographic interpretation of entanglement entropy in conformal field theories dual to supergravity on anti-de Sitter space is generalized to include entanglement entropy of black holes living on the boundary of anti-de Sitter space. The generalized proposal is verified in boundary dimensions d=2 and d=4 for both the uv-divergent and uv-finite terms. In dimension d=4 an expansion of entanglement entropy in terms of size L of the subsystem outside the black hole is considered. A new term in the entropy of dual strongly coupled conformal-field theory, which universally grows as L(2)lnL and is proportional to the value of the obstruction tensor at the black hole horizon, is predicted.
Abstract The odd entanglement entropy (OEE) for bipartite states in a class of (1+1)(1+1)(1+1) ( 1 + 1 ) -dimensional Galilean conformal field theories ( GCFT1+1GCFT_{1+1}GCFT1+1​ G C F T 1 + 1 ) is obtained through an appropriate replica technique. In this context our results are compared with the entanglement wedge cross section (EWCS) for (2+1)(2+1)(2+1) ( 2 + 1 ) -dimensional asymptotically flat geometries dual to the GCFT1+1GCFT_{1+1}GCFT1+1​ G C F T 1 + 1 in the framework of flat holography. We find that our results are consistent with the duality of the difference between the odd entanglement entropy and the entanglement entropy of bipartite states, with the bulk EWCS for flat holographic scenarios.
A bstract We derive dynamics of the entanglement wedge cross section from the reflected entropy for local operator quench states in the holographic CFT. By comparing between the reflected entropy and the mutual information in this dynamical setup, we argue that (1) the reflected entropy can diagnose a new perspective of the chaotic nature for given mixed states and (2) it can also characterize classical correlations in the subregion/subregion duality. Moreover, we point out that we must improve the bulk interpretation of a heavy state even in the case of well-studied entanglement entropy. Finally, we show that we can derive the same results from the odd entanglement entropy. The present paper is an extended version of our earlier report arXiv:1907.06646 and includes many new results: non-perturbative quantum correction to the reflected/odd entropy, detailed analysis in both CFT and bulk sides, many technical aspects of replica trick for reflected entropy which turn out to be important for general setup, and explicit forms of multi-point semi- classical conformal blocks under consideration.
It was proposed by Ryu and Takayanagi that the entanglement entropy in conformal field theory (CFT) is related through the AdS/CFT correspondence to the area of a minimal surface in the bulk. We apply this holographic geometrical method of calculating the entanglement entropy to study the vacuum case of a CFT which is holographically dual to empty anti-de Sitter (AdS) spacetime. We present all possible minimal surfaces spanned on one or two spherical boundaries at AdS infinity. We give exact analytical expressions for the regularized areas of these surfaces and identify finite renormalized quantities. In the case of two disjoint boundaries the existence of two different phases of the entanglement entropy is confirmed. A trivial phase corresponds to two disconnected minimal surfaces, while the other one corresponds to a tube connecting the spherical boundaries. A transition between these phases is reminiscent of the finite temperature deconfinement transition in the CFT on the boundary. The exact analytical results are thus consistent with previous numerical and approximate computations. We also briefly discuss the character of a spacetime extension of the minimal surface spanned on two uniformly accelerated boundaries.