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”

“The RT proposal has been extended to nonstatic geometries in [10].”

“There is a generalization of the Ryu-Takayanagi conjecture for situations that are time dependent [34].”

“However, for general dynamic spacetimes [27], and taking into account quantum corrections [28, 29], it is the quantum extremal surface, homologous to B, with the smallest generalised entropy A(χB)/4GN + Sbulk(χB).”

“Our ideal candidate for a formula for holographic entanglement entropy in a nonclassical bulk should therefore (1) be sensitive to quantum effects at any order in ~, (2) reproduce the classical HRT formula at O(~−1), and (3) reproduce the result of FLM at O(~0).”

“The five-qubit code is a [[5, 1, 3]]2 perfect code with distance 3 encoding one logical qubit in five physical qubits.”

“This resonates with the longstanding suspicion that some small corrections to the Hawking calculation (of order e−SBH) could be responsible for making black (1)For example [4, 5, 6, 7, 8, 9]; for a review see [10].”

“ted with a given region Ain the dual holographic eld theory [1{3]. At the level of classical bulk physics, W A is obtained by rst constructing the associated Hubeny-Rangamani-Takayanagi (HRT) surface [4], which is the minimal-area codimension-2 bulk extremal surface homologous to Ain an { 1 { appropriate sense [5]. Th…”

“This formula, for example, when applied to rotating BTZ black holes correctly reproduces the entanglement entropy expected from CFT2 [43].”

“The conjectured holographic recipe for computing this entropy is the following [1, 2].”

“In this case, the prescription of extremizing the generalized entropy in [7] is equivalent, at leading order, to the standard RT/HRT prescription [5, 6] of extremizing the area.”

“The construction involved connecting the boundaries by (D − 2)-dimensional surfaces similar to the Ryu-Takayanagi surfaces [16][17] used by Hartman and Maldacena”

“If we know the state on any spatial region, we also know it in the causal diamond associated to that region, recall figure 6.”

“A leading order holographic prescription for computing entanglement entropy was proposed in [21, 22] and it was extended to the next order in GN in [23] (see also [24]).”

“A leading order holographic prescription for computing entanglement entropy was proposed in [21, 22] and it was extended to the next order in GN in [23] (see also [24]).”

“We can define the spin-down region in such a minimal energy configuration as EA, which bounds the boundary region A, and corresponds to the region known as the entanglement wedge in the literature [23, 24].”

“This is the proposal of Ryu and Takayanagi (RT), later generalized by Hubeny, Rangamani, and Takayanagi (HRT), for a holographic expression for the von Neumann entropy of a subregion in the boundary theory [91, 92].”

“This shock disrupts the Ryu Takayanagi surface [11, 12] passing through the wormhole [13, 14].”

“A second one is a (1)Generalizations of the proposal of [1] to time dependent states were discussed in [6].”

“Physical review letters, 55(26):2924, 1985.”

“, [1, 2, 25, 32–38]), it is given by extremizing a certain functional of the bulk metric over codimension-two bulk surfaces whose boundary coincides with ∂A in the boundary CFT.”

“Along the way, we will also slightly sharpen the statement of the HRT formula, and in particular clarify the homology condition on EA.”

“3 Strictly speaking, we need to apply the covariant prescription of holographic entanglement entropy [38] in such a Lorentzian time-dependent spacetime.”

“The most common situation studied so far concerns a sudden quench of some coupling of the model Hamiltonian [6]– [10].”

“Moreover, this reconstruction can be extended [12,15–18] into the larger entanglement wedge of A, denoted EA and defined as the bulk domain of dependence of any bulk achronal surface whose only boundaries are A and the Hubeny/Rangamani/Takayanagi (HRT) surface γA associated to A [19].”

“16) is stationary (but not static), we must apply the covariant prescription suggested by [62] to evaluate the holographic entanglement entropy.”

“As we review below, the left side may be calculated using the Ryu-Takayanagi proposal [9, 10], while the right side can be calculated from the asymptotic form of the metric.”

“vour of quasi-local horizons comes from the analysis of holographic entanglement entropy (for specified regions on the boundary) in time dependent states of the dual field theory, see the discussion in [99]. 40As explained in [60] we use the term apparent horizon to denote a co-dimension one surface in the spacetime, wh…”

“tempts to prove the formula include [16,17]. One-loop corrections { 4 { were analyzed in [18,19]. A covariant version of the prescription which applies in general time-dependent cases was proposed in [20]. Let us review the derivation of the Ryu-Takayanagi prescription. The basic idea is to use the replica trick and e…”

“In this covariant setup, the holographic entanglement entropy [3] is given by”

“Here we work in the Euclidean signature as in [10] by assuming that the entangling region is at a moment of time-reflection symmetry, and leave covariant generalizations along the directions of [21, 22] to future work.”

“4) holds, interpreting γ as an extremal surface in the unrestricted gravity background subject to the same boundary conditions [19].”

“Hence, more generally, it would be useful to develop a better understanding of the geometry of the extremal bulk surface appearing in the calculation of holographic EE, perhaps along the lines of [51].”

“lations of S rad using the gravitational path integral (and in particular that this was implicit in earlier derivations of the quantum corrected Ryu-Takayanagi [50,51] and Hubeny-Rangamani-Takayanagi [52] entropy formulae [53,54]). While at this level the physical mechanisms behind such results remain somewhat mysteri…”

“A natural extension of this prescription to a time-dependent bulk spacetime refers to the Hubeny-Rangamani-Takayanagi (HRT) prescription for holographic entanglement entropy [36] (see also [37]) as follows:7”

“1 One strong evidence for this correspondence between holography and tensor networks, apart from the symmetry considerations, is the fact that the holographic entanglement entropy formula [15, 16] can naturally be explained in this approach by counting the number of entangling links in the networks.”

“Following this, assuming unbroken replica symmetry, we will argue that the entanglement entropy of the said density matrix is computed in terms of the area functional on a codimension-2 extremal surface in the geometry, thus deriving the HRT prescription [3].”

“Indeed, following the covariant conjecture of holographic entanglement entropy [8] in order to define the corresponding holographic complexity one should compute the volume of a part of the space time enclosed by the extremal co-dimension two hypersurface appearing in the computation of the covariant holographic entan…”

“This is precisely what appears in the leading-order formula for the entanglement entropy of one of the boundaries [65, 66]; this suggests a possibly deep connection between the RyuTakayanagi/Hubeny-Rangamani-Takayanagi formula and the factorization problem.”

“This agrees with the result of [27] obtained via the replica trick and holographic calculations.”

“There, prescriptions for the holographic dual of the entanglement entropy exist [31], and there is a conjecture for a good notion of Noether entropy in time dependent setups in any theory of gravity [9].”

“And presumably also the HRT formula [63], if one analytically continues using a complexified manifold.”

“These concepts gained a wider scope in the context of the AdS/CFT correspondence, where the Ryu-Takayanagi (RT) prescription [6–9] applies the same geometric expression to extremal bulk surfaces in evaluating the entanglement entropy for generic subregions on the boundary theory.”

“The entanglement entropy between these regions, S(R), can be computed using – 78 – the standard RT/HRT formula in the bulk.”

“The entanglement entropy between these regions, S(R), can be computed using – 78 – the standard RT/HRT formula in the bulk.”

“1 Introduction The RT/HRT/EW formula [1–3] for computing entanglement entropies is a remarkable entry in the holographic dictionary.”

“The topological condition of γA of HEE requires that the subsystem A should be homologous to γA.”

“5) matches the entanglement entropy for a CFT on a line at inverse temperature β and angular potential Ω [14].”

“In this section we make a connection between our results and the recently proposed JLMS formula [6]: KCFTA = Area∂M\A 4GN +KbulkM + · · ·+O(GN ) (4.1) where A denotes the boundary subsystem, and M denotes the bulk region enclosed by the minimal/extremal Ryu-Takayanagi/HRT [54, 55] surface ∂M\A (which ends on the – 17 –”

“Inspired by the AdS/CFT correspondence of string theory, Ryu and Takayanagi have proposed an alternative “holographic” approach to defining and regulating entanglement entropy [120, 121].”

“One very important property of the AdS/CFT correspondence is the Ryu-Takayanagi (RT) formula [8–13].”

“The covariant generalization of RT, the Hubeny–Rangamani–Takayanagi (HRT) formula [3], replaces the minimal surface with an extremal (spacelike codimension-2) surface.”

“The new approach builds on insights coming from holographic entanglement entropy [16–19] and its extension to include quantum contributions [20, 21].”

“At this time, the geometry that the HRT surface is experiencing is that of a static black hole.”

“Because all subsequent higher-partition entanglement information is encoded in the mutual information between all suitably chosen partitions, the approximate geometric information above the chosen scale of partitioning can be characterized by the graph representation.”

“These features follow from the fact that the holographic entanglement entropy [89, 90] (see, e.”

“2) Because the time-symmetric surface is precisely the Riemann surface Σ, the corresponding entanglement entropies can be obtained by the Ryu-Takayanagi formula from the minimal geodesics in Σ [5, 22].”

“4 Generalization to the covariant case is discussed in [55].”

“However, in general time-dependent backgrounds we can employ the covariant formulation in [37].”

“According to the prescription of the minimal quantum extremal surface [9–11], states in some regions in R, which are called islands I(⊂ R), should be excluded from the states to be traced out.”

“This problem was also discussed in [11].”

“See [12–27] for selected pre-cursors and elaborations on the connection between gravity and entanglement.”

“xiliary radiation system coupled to a system that would otherwise describe an equilibrium black hole2. The new insights come by making use of the quantum version [18,19] of the Ryu-Takayanagi formula [20,21], which gives the gravity interpretation of entanglement entropies for subsystems of a holographic quantum syste…”

“18); as described in [4] this can be obtained efficiently by passing to new coordinate where the metric (3.”