Abstract
We propose a holographic entanglement negativity conjecture involving the bulk geometry, for mixed states of adjacent intervals in (1+1)-dimensional dual conformal field theories through the AdS/CFT correspondence. The holographic entanglement negativity is obtained from a specific algebraic sum of the geodesics anchored on respective intervals on the boundary which reduces to the holographic mutual information between them. Utilizing our conjecture we obtain the entanglement negativity of adjacent intervals in zero and finite temperature (1+1)-dimensional holographic conformal field theories dual to the bulk AdS3 vacuum and the Euclidean BTZ black hole respectively. Our holographic conjecture exactly reproduces the conformal field theory results obtained through the replica technique, in the large central charge limit. We briefly elucidate the corresponding issue for the AdSd+1/CFTd scenario.
Highlights
Quantum entanglement in recent times has impacted an expansive list of theoretical issues from condensed matter physics to quantum gravity through the holographic AdS/CF T correspondence [1,2,3,4,5]
For the AdS3/CF T2 scenario the static minimal surface reduces to a space like geodesic in the bulk AdS3 geometry anchored on the appropriate spatial interval in the dual CF T1+1
To summarize we have established a holographic entanglement negativity conjecture for mixed states of adjacent intervals in zero and finite temperature dual CF T1+1 in the AdS3/CF T2 scenario. Note that this configuration is a mixed state as the degrees of freedom of the rest of the system are traced over and the corresponding entanglement negativity characterizes the entanglement between the adjacent intervals
Summary
Quantum entanglement in recent times has impacted an expansive list of theoretical issues from condensed matter physics to quantum gravity through the holographic AdS/CF T correspondence [1,2,3,4,5]. Following [6], in a seminal work Ryu and Takayanagi proposed a holographic characterization of the entanglement entropy in d-dimensional conformal field theories (CF Td), involving bulk dual AdSd+1 geometries through the AdS/CF T correspondence [8, 9] ( for an extensive review see [10]). In a recent interesting communication two of the present authors (VM and GS) in the collaborations [17,18,19](CMS), proposed a holographic conjecture for the entanglement negativity of mixed states in holographic CF Tds. The conjecture involves a specific algebraic sum of the areas of co-dimension two bulk extremal surfaces ( geodesic lengths in AdS3/CF T2 ) anchored on the corresponding subsystems. We summarize our results in the last section IV
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