A bstract In this work, we develop a systematic formalism to evaluate the upper bound of a large family of holographic entanglement entropy combinations when fixing n subsystems and fine-tuning one other subsystem. The upper bound configurations and values of these entropy combinations can be derived and classified. The upper bound of these entropy combinations reveals holographic n + 1-partite entanglement that n fixed subsystems participate in. In AdS 3 /CFT 2 , AdS 4 /CFT 3 , and even higher-dimensional holography, one can, in principle, find different formulas of upper bound values, reflecting the fundamental difference in entanglement structure in different dimensions.
A bstract We show that there exists a huge amount of multipartite entanglement in holography by studying the upper bound for holographic n -partite information I n that n − 1 fixed boundary subregions participate. We develop methods to find the n -th region E that makes I n reach the upper bound. Through the explicit evaluation, it is shown that I n , an IR term without UV divergence, could diverge when the number of intervals or strips in region E approaches infinity. At this upper bound configuration, we could argue that I n fully comes from the n -partite global quantum entanglement. Our results indicate: fewer-partite entanglement in holography emerges from more-partite entanglement; n − 1 distant local subregions are highly n -partite entangling. Moreover, the relationship between the convexity of a boundary subregion and the multipartite entanglement it participates, and the difference between multipartite entanglement structure in different dimensions are revealed as well.
To analyze the holographic multipartite entanglement structure, we study the upper bound for holographic
n-partite information
(−1)nIn that
n−1 fixed boundary subregions participate together with an arbitrary region
E. In general cases, we could find regions
E that make
In approach the upper bound. For
n=3, we show that the upper bound of
−I3 is given by a quantity that we name the entanglement of state-constrained purification
EoSP(A:B). For
n≥4, we find that the upper bound of
In is finite in holographic CFT
1+1 but has UV divergences in higher dimensions, which reveals a fundamental difference in the entanglement structure in different dimensions. When
(−1)nIn reaches the information-theoretical upper bound, we argue that ( I_n ) fully accounts for multipartite global entanglement in these upper bound critical points, in contrast to usual cases where
In is not a perfect measure for multipartite entanglement. We further show that these results suggest that fewer-partite entanglement fully emerges from more-partite entanglement, and any
n−1 distant regions are fully
n-partite entangling in higher dimensions.
A bstract In this paper, we systematically study the measures of multi-partite entanglement with the aim of constructing those measures that can be computed in probe approximation in the holographic dual. We classify and count general measures as invariants of local unitary transformations. After formulating these measures in terms of permutation group elements, we derive conditions that a probe measure should satisfy and find a large class of solutions. These solutions are generalizations of the multi-entropy introduced in [1]. We derive their holographic dual with the assumption that the replica symmetry is unbroken in the bulk and check our prescription with explicit computations in 2 d CFTs. Analogous to the multi-entropy, the holographic dual of these measures is given by the weighted area of the minimal brane-web but with branes having differing tensions. We discuss the replica symmetry assumption and also how the already known entanglement measures, such as entanglement negativity and reflected entropy fit in our framework.
Entanglement entropy for a spatial partition of a quantum system is studied in theories which admit a dual description in terms of the anti-de Sitter (AdS) gravity one dimension higher. A general proof of the holographic formula which relates the entropy to the area of a codimension 2 minimal hypersurface embedded in the bulk AdS space is given. The entanglement entropy is determined by a partition function which is defined as a path integral over Riemannian AdS geometries with non-trivial boundary conditions. The topology of the Riemannian spaces puts restrictions on the choice of the minimal hypersurface for a given boundary conditions. The entanglement entropy is also considered in Randall-Sundrum braneworld models where its asymptotic expansion is derived when the curvature radius of the brane is much larger than the AdS radius. Special attention is payed to the geometrical structure of anomalous terms in the entropy in four dimensions. Modification of the holographic formula by the higher curvature terms in the bulk is briefly discussed.
This is an extended version of our short report hep-th/0603001, where a holographic interpretation of entanglement entropy in conformal field theories is proposed from AdS/CFT correspondence. In addition to a concise review of relevant recent progresses of entanglement entropy and details omitted in the earlier letter, this paper includes the following several new results: We give a more direct derivation of our claim which relates the entanglement entropy with the minimal area surfaces in the AdS3/CFT2 case as well as some further discussions on higher dimensional cases. Also the relation between the entanglement entropy and central charges in 4D conformal field theories is examined. We check that the logarithmic part of the 4D entanglement entropy computed in the CFT side agrees with the AdS5 result at least under a specific condition. Finally we estimate the entanglement entropy of massive theories in generic dimensions by making use of our proposal.
A bstract We consider holographic entanglement entropy in AdS black hole backgrounds by using the limit of large number of dimensions. By dividing the geometry to two patches (with one patch covering the vicinity of the black hole horizon and another covering the other regions), we are able to obtain fully analytic expressions for the entropy when the entanglement region is a strip. We argue that apart from conformal field theories at finite temperature in high number of dimensions, our method works for nearly critical theories in 3+1 dimensions. In the case of extremal black holes, dual to quantum critical systems, the results take a particularly simple form. We also comment on the case of soliton geometries. Finally, we analyze entanglement entropy for wide strips, and propose a general formula for the first subleading term in the expansion of the entropy in (inverse) system size for generic entanglement regions.