A covariant holographic entanglement entropy proposal

“, GKP-W relation [38, 39] for the AdS/CFT correspondence.”

“, GKP-W relation [38, 39] for the AdS/CFT correspondence.”

“In string theory a natural realization of the holographic principle is manifested by the AdS/CFT correspondence [18, 19] which gives us a precise map between a quantum gravity theory on an asymptotically AdS spacetime M and an ordinary QFT on the conformal boundary ∂M of M.”

“to the bulk-boundary relation within the AdS/CFT context in § 4. In § 5 we illustrate the calculations of the entanglement entropy using our covariant proposal and demonstrate the consistent agreement with the minimal surface prescri pt ion of [24, 10].”

“Roughly speaking, the holographic principle states that the number of degrees of freedom in a quantum theory of gravity scales with the area of the system, in contrast to standard QFTs where the entropy is extensive and scales with the volume [15, 16, 17].”

“Recently, entanglement entropies of various 1+1 and 2+1 dimensional condensed matter systems have been actively investigated in order to understand zero temperature quantum phase transitions [11, 4, 12, 13, 14].”

“Using the relation between the central charge of the dual CFT and the bulk Newton’s constant c = 3 2G (3) N [50], we obtain the expression in the dual CFT language”

“In dynamical situations, however, one needs to upgrade the recipe and use the covariant HRRT formula instead [3].”

“A precise prescription for computing entanglement entropy for strongly coupled field theories with AdS duals was proposed in [31] and later generalized in [32] (for a good review see [33]).”

“11 Here we will no longer assume that the geometry is static, so EA is now constructed with the Hubeny–Rangamani–Takayanagi [6] surface associated with A, which we will still denote A.”

“where γ is the quantum extremal surface of smallest generalized entropy [8, 15–17], there is some freedom in whether we view their microstates as contributing to the area term or the bulk entropy term.”

“Indeed, the geometrization of the notions of entanglement entropy in the gauge/gravity context for holographic field theories as originally proposed in [9, 10] (and made geometrically covariant in [11]) makes one wonder if there are further (1)LOCC for two parties consists of steps in each of which any party is allowe…”

“If only chiral primary operators (and their descendants) are kept in this expansion, the resulting EE coincides with the one evaluated at the gravity point in the CFT moduli space, which, on the other hand, can be holographically computed via the Ryu-Takayanagi formula [17] (and its generalizations [20]).”

“One can study the entropy of the black hole or its complementary subsystem (containing the Hawking radiation) as a function of time, using the Engelhardt-Wall prescription [11] (see also [12]) for calculating von Neumann entropy — a generalization of the Hubeny-Ryu-Rangamani-Takayanagi (HRT) prescription [13, 14] to i…”

“1), the increase EE of a single interval of length l [23] is given by”

“It was shown in [4] that the HRT formula, applied to the BTZ spacetime, reproduces (4.”

“Holographic entanglement entropy was first considered in dynamical situations by Hubeny, Rangamani, and Takayanagi (HRT) [17].”

“However, it is known [21, 27, 28] that there can be discontinuities in the position of the HRT surface [46], and consequently in the entanglement wedge, when the time t is of the order of the size of the boundary subregion.”

“This conjectured correspondence is made precise for static spacetimes by the Ryu-Takayanagi (RT) formula [2, 3], and for dynamical spacetimes by the Hubeny-Rangamani-Takayanagi (HRT) formula [4].”

“Fursaev suggested a derivation [11] of the Ryu-Takayanagi (1)A covariant version was proposed in [5].”

“For this we need to develop the replica trick in real times and presumably this will have some bearing on the HRT conjecture [61] for how to generalize Ryu-Takayanagi to real times.”

“Entanglement entropy plays important roles in various fields of quantum physics including string theory [1–16], condensed matter physics [17–22], and the physics of the black hole [23–27].”

“tion of the CFT to the generalized entropy is calculated by nding extremal HRT surfaces, i.e., geodesics, in the AdS 3 bulk, in accord with the usual prescription for holographic entanglement entropy [39,40]. In general, these geodesics may connect that endpoints of the relevant intervals in the boundary theory, howev…”

“One interesting quantity to consider in such studies is the holographic entanglement entropy [8, 9], which was generalized to time-dependent backgrounds in [10], in particular the collapse of a homogeneous massless shell in the AdS space-time, leading to black hole formation in the bulk, which corresponds to thermaliz…”

“hat disrupts this mutual information, we will see the wormhole geometry grow longer. The computation of mutual information is made possible by the Ryu-Takayanagi prescription for entanglement entropy [7, 8]. To a region Aon the left asymptotic boundary we associate 3 a minimal surface in the bulk whose boundary coinci…”

“In fact, it has been argued previously in several different contexts [24, 25] that it may be more appropriate to associate the entropy of the CFT configuration to the area of the apparent horizon rather than the event horizon in the bulk dual.”

“In the AdS/CFT correspondence [1], the holographic entanglement entropy formula [2, 3] relates the entanglement structure of the CFT with the geometrical structure of the dual spacetime.”

“As shown in [1, 2], by applying the standard Ryu-Takayanagi (RT) formula [14] with its generalization including time dependence [15] and quantum correction [16, 17], one finds that the minimal quantum extremal surface has a phase transition at the Page time, which then gives the entropy curve for the black hole in acc…”

“In time-dependent backgrounds, the holographic entanglement entropy is given by the extremal surface area [9].”

“for purposes of holographic entanglement entropy [12, 13]), we can prove the existence of barriers in more circumstances.”

“We can calculate SA(t) via AdS/CFT by applying the holographic formula proposed in [14], which generalizes the holographic formula in static spacetime found in [13].”

“For a subregion A on the boundary, its holographic entanglement entropy (HEE) is captured by a codimensiontwo bulk surface with vanishing expansion of geodesics [32], i.”

“Associated to the region A on the boundary there is a bulk region a between A and the associated HRT extremal surface mA [6, 33].”

“Although this island prescription is first suggested in the context of the holography (or AdS/CFT correspondence) by Ryu and Takayanagi [19] and its further developments [6, 9–11, 16, 20–22], it can be applied to any quantum system coupled to gravity.”

“2) Of course since the metric is not static one needs to use the covariant proposal for the holographic entanglement entropy [5].”

“There is no obvious reason that a geometric region on the boundary has to correspond to a geometric region in the bulk, but there are strong arguments for such a subregion duality in certain simple cases [3], and intriguing hints [4–6] that it may be true more generally.”

“eralization should be guided by the covariant version of the Ryu-Takayanagi proposal [68], which also satisfies the strong subadditivity of entropy [69].”

“In particular, the holographic formula relating entanglement entropy to bulk area of a boundary homologous minimal surface by Ryu and Takayanagi [3, 4] (later extended to extremal surfaces in the covariant case by [5]) has spurred a great deal of interest, from its ability to constrain what sets of states can be dual…”

“(14) The holographic EoP is computed from the length of the entanglement wedge cross section ΣminAB in the hyperbolic space ds2 = (dx2 + dz2)/z2, which is a time slice of Poincare AdS3.”

“Such surfaces play a central role in the quantum-corrected [20,21] Ryu-Takayanagi (RT) [22]/ Hubeny-Rangamani-Takayanagi [23] prescription, which”

“Furthermore, perhaps EW surfaces anchored to boundary-anchored HRT surfaces [19] that probe behind the event horizon of black holes formed from collapse [20] have areas which can be calculated both holographically and directly in the boundary field theory, providing a nontrivial check of the smoothness of the region b…”

“We can also extend the holographic entanglement entropy [5, 6, 92] to AdS/BCFT setups.”

“r the Renyi entropy of a thermal ensemble in a 2d CFT in the large climit,8 trˆn ;Cx = sinh ˇL(1 x) n ; trˆn vac;Cx = sin ˇ2x L n n= c 12 n 1 n (2.20) 8We use the holographic expressions [4] for thermal Renyi entropies, though we expect that our result to be be unchanged if we pick a dierent sort of CFT (analogous to…”

“The QES formula is a generalization of the (classical) Ryu– Takayanagi formula [34, 35], which says that the von Neumann entropy in quantum gravity SvN(ΣX) of a codimension-1 slice ΣX bounded by a QES X may be computed in the semi-classical approximation using the following extremization prescription”

“Unlike AdS where specifying boundary data fixes the extremization problem, dS extremal surfaces starting at late times on I+ do not return, thus requiring extra data on boundary conditions in the far past.”

“In the context of holographic entanglement entropy, maximal volume slices were initially considered—but ruled out—in [21] as a step in covariantizing the Ryu-Takayanagi proposal.”

“The Ryu-Takayanagi (RT) formula [2] and its generalizations [3–5] provide important clues about how the bulk degrees of freedom corresponds to the boundary ones.”

“n Neumann and Renyi entropies of this nal state should satisfy the unitarity constraint (1.2). In holography, the von Neumann entropy S(A) 1 for a subregion Ais found from the area of the HRT surface [63,64]. For a black hole formed from collapse, the HRT surface for A is the same as that for A, and thus the unitarity…”

“Although the island formula is applicable to generic spacetimes, it was inspired by the Ryu-Takayanagi (RT)/Hubeny-Rangamani-Takayanagi (HRT) formula for computing the entanglement entropy of a subsystem in a holographic CFTd [34–37], specifically its quantum corrected formula proposed in [41, 42] .”

“When we freeze gravity in the interior of a piece of the static patch (and a similar antipodal region), we will show that generalized entropy computations in de Sitter space have a natural quantum extremal surface [7, 33], stemming from the classical extremal surface (the bifurcation surface) [34, 35], which is a mini…”

“As discussed in [5], for general non-static situations one needs to replace the minimal surface by an extremal surface in the bulk.”

“This suggests that at least caution should be taken in applying RT (or HRT) proposal for holography beyond AdS/CFT. Generalization to general temperatures can be done by considering the further mapping from BTZ black holes to Poincaré AdS3 [73], and then follow the steps of section 4.3 in the main text, or section 5.…”

“1) in a large class of CFTs, there is always an associated classical geometry (defined to second order in perturbation theory) which captures the entanglement entropies of ball-shaped regions via the Ryu-Takayanagi formula [9, 10] and which satisfies Einstein’s equations perturbatively to second order.”

“Finally, we compute the entanglement entropy and the mutual information using the holographic prescription [28, 29].”

“The formula was first proposed for stationary asymptotic anti-de-Sitter (AdS) spacetime, and has been subsequently generalized to time-dependent cases, known as the Hubeny-Rangamani-Ryu-Takayanagi (HRRT) formula [6].”

“In particular, it has recently become a standard term in the high-energy physics vocabulary due mainly to the AdS/CFT correspondence, the Ryu-Takayanagi formula [1] and its generalization [2].”

“The Hubeny-Rangamani-Takayanagi (HRT) prescription gives the covariant generalization of RT that applies to arbitrary dynamical spacetimes [7, 8].”

“[6].”

“There are numerous extensions of the formula including time-dependent theories [10, 11], quantum corrections [12], and Rényi entropies [13].”

“15There is a dynamical version of the holographic entropy, given by [49]: SHRT[R] = min extγArea[γ]/4GN .”

“It would be interesting to explore real-time holography in the context of entanglement entropy, using the techniques of [29] for the HRT functional [45].”

“The island formula somehow stems from the investigations of the quantum corrections [19–21] of the Ryu-Takayanagi (RT) formula [22, 23].”