Abstract Using the Hubeny-Rangamani-Takayanagi (HRT) conjectured formula for entanglement entropy in the context of the AdS/CFT correspondence with time-dependent backgrounds, we investigate the relation between the bulk null energy condition (NEC) of the stress-energy tensor with the strong sub-additivity (SSA) property of entanglement entropy in the boundary theory. In a background that interpolates between an AdS to an AdS-Reissner-Nordstrom-type geometry, we find that generically there always exists a critical surface beyond which the violation of NEC would naively occur. However, the extremal area surfaces that determine the entanglement entropy for the boundary theory, can penetrate into this forbidden region only for certain choices for the mass and the charge functions in the background. This penetration is then perceived as the violation of SSA in the boundary theory. We also find that this happens only when the critical surface lies above the apparent horizon, but not otherwise. We conjecture that SSA, which is thus non- trivially related to NEC, also characterizes the entire time-evolution process along which the dual field theory may thermalize.
Abstract Using the Hubeny-Rangamani-Takayanagi (HRT) conjectured formula for entanglement entropy in the context of the AdS/CFT correspondence with time-dependent backgrounds, we investigate the relation between the bulk null energy condition (NEC) of the stress-energy tensor with the strong sub-additivity (SSA) property of entanglement entropy in the boundary theory. In a background that interpolates between an AdS to an AdS-Reissner-Nordstrom-type geometry, we find that generically there always exists a critical surface beyond which the violation of NEC would naively occur. However, the extremal area surfaces that determine the entanglement entropy for the boundary theory, can penetrate into this forbidden region only for certain choices for the mass and the charge functions in the background. This penetration is then perceived as the violation of SSA in the boundary theory. We also find that this happens only when the critical surface lies above the apparent horizon, but not otherwise. We conjecture that SSA, which is thus non- trivially related to NEC, also characterizes the entire time-evolution process along which the dual field theory may thermalize.
Abstract: \nTensor networks,ToverlineTTT¯, and broader notions of a holographic principle all motivate the idea that some notion of gravitational holography should persist in the presence of a radial cutoff. But in the absence of time-reflection symmetry, the areas of Hubeny-Rangamani-Takayanagi surfaces anchored to the radial cutoff generally violate strong sub-additivity, even when the associated boundary regions are spacelike separated as defined by both bulk and boundary notions of causality. We thus propose an alternate definition of cutoff-holographic entropy using a restricted maximin prescription anchored to a codimension 2 cutoff surface. For bulk solutions that respect the null energy condition, we show that the resulting areas satisfy SSA, entanglement wedge nesting, and monogamy of mutual information in parallel with cutoff free results in AdS. These results hold even when the cutoff surface fails to be convex.
One decade ago, Ryu-Takayanagi explicitly introduced a formula that relates the entropy of a subregion in CFT system to a geometrical quantity which is called minimal surface in hyperbolic space. This formula extended to the idea of connection of gravy to quantum mechanics of gauge/gravity duality. This duality which can help us to learn a more interesting feature of each side from the other. Quantum systems obey from some constraints come from the quantum information theory. I would be interesting to find out what is the dual of this constraint in the gravitation system. Dual to the specific class of quantum theories which is called conformal field theories. One of the most significant constraint that QFTs should obey is the strong subadditivity of entanglement entropy. These constraints let the theories have bound on the energy spectrum from the below; Recently there has been the development that the combination of monotonicity of relative entropy and the strong subadditivity of entanglement entropy is equal to have a specific bound on the energy momentum tensor, called quantum null energy condition. In this thesis, we re-look to this argument by introducing the entanglement density and obtain a differential operator from the strong subadditivity and exploiting from the Markov property of the vacuum of CFT. In the next step, by using from the Ryu-Takayangi, we rewrite the strong subadditivity inequality regarding geometrical quantities. By using from the kinematic languages and intertwinement, we realize that the strong subadditivity at the boundary implies new bound on averaged energy condition which has some common feature with the quantum null energy condition statement.
With an aim towards understanding the time-dependence of entanglement entropy in generic quantum field theories, we propose a covariant generalization of the holographic entanglement entropy proposal of hep-th/0603001. Apart from providing several examples of possible covariant generalizations, we study a particular construction based on light-sheets, motivated in similar spirit to the covariant entropy bound underlying the holographic principle. In particular, we argue that the entanglement entropy associated with a specified region on the boundary in the context of the AdS/CFT correspondence is given by the area of a co-dimension two bulk surface with vanishing expansions of null geodesics. We demonstrate our construction with several examples to illustrate its reduction to the holographic entanglement entropy proposal in static spacetimes. We further show how this proposal may be used to understand the time evolution of entanglement entropy in a time varying QFT state dual to a collapsing black hole background. Finally, we use our proposal to argue that the Euclidean wormhole geometries with multiple boundaries should be regarded as states in a non-interacting but entangled set of QFTs, one associated to each boundary.
Abstract In holographic duality, the entanglement entropy of a boundary region is proposed to be dual to the area of an extremal codimension-2 surface that is homologous to the boundary region, known as the Hubeny-Rangamani-Takayanagi (HRT) surface. In this paper, we study when the HRT surfaces of two boundary subregions R, A are in the same Cauchy slice. This condition is necessary for the subregion-subregion mapping to be local for both subregions and for states to have a tensor network description. To quantify this, we study the area of a surface that is homologous to A and is extremal except at possible intersections with the HRT surface of R (minimizing over all such possible surfaces), which we call the constrained area. We give a boundary proposal for an upper bound of this quantity, a bound which is saturated when the constrained surface intersects the HRT surface of R at a constant angle. This boundary quantity is the minimum entropy of region A in a modular evolved state — a state that has been evolved unitarily with the modular Hamiltonian of R. We can prove this formula in two boundary dimensions or when the modular Hamiltonian is local. This modular minimal entropy is a boundary quantity that probes bulk causality and, from this quantity, we can extract whether two HRT surfaces are in the future or past of each other. These entropies satisfy some inequalities reminiscent of strong subadditivity and can be used to remove certain corner divergences.
We study holographic entanglement entropy in 3-dimensional AdS gravity with conformal boundary conditions which fix the conformal class of the boundary metric and its extrinsic curvature, K, while leaving the Weyl mode dynamical. Extending Lewkowycz-Maldacena-Dong's replica construction, we derive the corresponding holographic entanglement entropy formula. We show that the fluctuating Weyl mode does not contribute additional entropy. The entropy of the full boundary is therefore the Bekenstein-Hawking entropy, and the entropy of a boundary subregion continues to obey the Ryu-Takayanagi prescription, namely, the area of a minimal surface divided by 4GN. We carry out explicit calculations for global AdS, rotating and non-rotating BTZ geometries. For an interval in AdS3 we find that the K-dependent holographic entanglement entropy is governed by cm=2GN3ℓ. For a thermal state at high conformal temperatures we find that, the entropy is governed by ceff=2GN3ℓ2Kℓ−K2ℓ2−4, the same effective central charge that governs the Cardy-like density of states, in agreement with previous results in the literature. Finally, we also compute the entanglement entropy directly from the conjectured dual boundary theory - a holographic CFT coupled with time-like Liouville theory and deformed by a marginal TTˉ like operator - and find SEE=3ceffln(ε2Rsinϕ0). This result for the state with no operator insertions (vacuum state ), provides an independent boundary realization of ceff while clarifying that this state is not the state dual to global AdS.
Holographic entanglement entropy of two disjoint intervals in AdS3/CFT2
The Ryu-Takayanagi conjecture predicts a holographic dual of the entanglement entropy of a CFT. It proposes that the entanglement entropy is given by the area of the minimal surface in the dual spacetime. In the semi-classical limit, this conjecture is supported by the saddle point approximation. If there are multiple classical solutions, it is assumed that only the minimal action contributes to the entanglement entropy. However, we will point out that these saddles equally contribute to the entanglement entropy in some cases. Therefore, the derivation of the conjecture is incomplete if there are multiple extremal surfaces that extend from a sub-system on the AdS boundary. We will consider two disjoint intervals in CFT1+1 as the simplest but non-trivial example, and propose another candidate for a holographic dual of the entanglement entropy of this system, which is the sum of all the signed areas of extremal surfaces in the dual spacetime. After that, we will derive it from the CFT calculations and propose the corresponding gravity side action.