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”

“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].”

“The essential idea behind the picture of [24, 10] is the following: by virtue of time translation invariance, the boundary spacelike foliation naturally extends into the bulk to provide a canonical spacelike foliation”

“Interestingly, for a long time it has been known that the leading ultraviolet divergent contribution to the entanglement entropy SA in QFTs is proportional to the area of the boundary ∂A of the subsystem A (known as the area law of entanglement entropy) [22, 23].”

“In a two dimensional CFT, we can analytically calculate the entanglement entropy for arbitrary choice of the subsystem At as shown recently in [5], generalizing the previously known result [6].”

“We argued that our covariant proposal can be derived naturally using the light-sheet construction and thus is closely related to the covariant entropy bound (Bousso bound) [35].”

“22 One could try to interpret this as unitary evolution in a tensor product theory, where the second Hilbert space is hidden behind the horizon, as in the eternal AdS black hole [21].”

“hat theleadingdivergence ofentanglement entropy scales astheareaoftheboundary ∂Abetween the two regions (or sub-systems, as they are conventionally referred to in the entanglement entropy literature) [22, 23]. The intuitive reason of this area law for the divergent part is that the most entangled degrees of freedom ar…”

“Explicit examples are borne out in the analysis of [34] where the authors analyse the situation in two dimensional field theories.”

“A dynamical horizon is defined to be a smooth, co-dimension one spacelike submanifold of the spacetime, which can be foliated by a family of closed spacelike surfaces, such that the leaves of the foliation have one null expansion vanishing and the other null expansion being strictly negative [55, 47] as in (6.”

“We argued that our covariant proposal can be derived naturally using the light-sheet construction and thus is closely related to the covariant entropy bound (Bousso bound) [35].”

“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].”

“Clearly there is a well-defined notion of the central charges for conformal field theories in d > 2 [2], which may quantify the total degrees of freedom.”

“We argued that our covariant proposal can be derived naturally using the light-sheet construction and thus is closely related to the covariant entropy bound (Bousso bound) [35].”

“Moreover, an analogue of the Zamolodchikov’s c-theorem (called entropic c-theorem) has been shown in [7, 8] (see also [9]).”

“These are the Euclidean AdS wormholes discussed in [40].”

“A dynamical horizon is defined to be a smooth, co-dimension one spacelike submanifold of the spacetime, which can be foliated by a family of closed spacelike surfaces, such that the leaves of the foliation have one null expansion vanishing and the other null expansion being strictly negative [55, 47] as in (6.”

“Interestingly, the existence of such a minimal surface at the throat also plays the crucial role when we present a generic definition of wormhole as discussed in [57].”

“, [63] for other recent interesting approach).”

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

“the integral of the entropy flux on the light-sheet [44]) is bounded by the area of spacelike surface as follows: SLS ≤ Area(S) 4 GN .”

“It would also be interesting to generalize the covariant holographic formula beyond the supergravity approximation assumed in the above discussion (see [27] for recent progress in the Euclidean case).”

“Refer to [4] for a short review of entanglement entropy in QFT.”

“This is because the emergence of the bubble means the disappearance of degrees of freedom as discussed in [3] about the static AdS bubble example.”

“Moreover, an analogue of the Zamolodchikov’s c-theorem (called entropic c-theorem) has been shown in [7, 8 ] (see also [9]).”

“3) 12 In this case we can easily prove the strong subadditivity of holographic entanglement entropy [25] as in [26] since two minimal surfaces on the same time slice can intersect with each other if they do so at the boundary of AdS.”

“However, even in such cases we expect to have an equivalent description in terms of a pure state by assuming another CFT sector hidden inside the horizons as in the Schwarzschild-AdS case [21], or other degrees of freedom in more general circumstances as in the examples of [41].”

“Our final example of a time-dependent asymptotically AdS background is the AdS bubble solution [59, 60] ds(2) = f(r) dχ(2) + dr(2) f(r) + r(2) ( −dτ 2 + cosh(2) τ dΩ(2)2 )”

“Moreover, this holographic relation is successfully applied to the brane-world black holes [28, 29] and de-Sitter spaces [30] as well, which enable us to interpret the horizon entropy with quantum corrections as the entanglement entropy (see also recent discussions [31, 32, 33]).”

“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].”

“Moreover, this holographic relation is successfully applied to the brane-world black holes [28, 29 ] and de-Sitter spaces [30] as well, which enable us to interpret the horizon entropy with quantum corrections as the entanglement entropy (see also recent discussions [31, 32, 33]).”

“A discussion of this entropy bound applied to the AdS/CFT, which stimulates our arguments below, can be found in [43].”

“ensity matrices ρ such that their trace is one i.e., Trρ = 1. In order to express SA(t) in the path integral formalism, we need to first describe the reduced density matrix in that formalism (see also [5, 34, 42, 10]). Taking the trace in the Hilbert space HB in (2.10) is equivalent to partially gluing two boundaries i…”

“Our final example of a time-dependent asymptotically AdS background is the AdS bubble solution [59, 60] ds(2) = f(r) dχ(2) + dr(2) f(r) + r(2) ( −dτ 2 + cosh(2) τ dΩ(2)2 )”

“Hence the metric extraction of [61] does not probe the bulk past null circular orbits.”

“Moreover, this holographic relation is successfully applied to the brane-world black holes [28, 29] and de-Sitter spaces [30] as well, which enable us to interpret the horizon entropy with quantum corrections as the entanglement entropy (see also recent discussions [31, 32, 33]).”

“Moreover, this holographic relation is successfully applied to the brane-world black holes [28, 29] and de-Sitter spaces [30] as well, which enable us to interpret the horizon entropy with quantum corrections as the entanglement entropy (see also recent discussions [31, 32, 33]).”

“Moreover, an analogue of the Zamolodchikov’s c-theorem (called entropic c-theorem) has been shown in [7, 8] (see also [9]).”

“Refer to [4] for a short review of entanglement entropy in QFT.”

“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).”