The first law of differential entropy and holographic complexity

Abstract

We construct the CFT dual of the first law of spherical causal diamonds in three-dimensional AdS spacetime. A spherically symmetric causal diamond in AdS3 is the domain of dependence of a spatial circular disk with vanishing extrinsic curvature. The bulk first law relates the variations of the area of the boundary of the disk, the spatial volume of the disk, the cosmological constant and the matter Hamiltonian. In this paper we specialize to first-order metric variations from pure AdS to the conical defect spacetime, and the bulk first law is derived following a coordinate based approach. The AdS/CFT dictionary connects the area of the boundary of the disk to the differential entropy in CFT2, and assuming the ‘complexity=volume’ conjecture, the volume of the disk is considered to be dual to the complexity of a cutoff CFT. On the CFT side we explicitly compute the differential entropy and holographic complexity for the vacuum state and the excited state dual to conical AdS using the kinematic space formalism. As a result, the boundary dual of the bulk first law relates the first-order variations of differential entropy and complexity to the variation of the scaling dimension of the excited state, which corresponds to the matter Hamiltonian variation in the bulk. We also include the variation of the central charge with associated chemical potential in the boundary first law. Finally, we comment on the boundary dual of the first law for the Wheeler-deWitt patch of AdS, and we propose an extension of our CFT first law to higher dimensions.

Quantum extremal islands made easy. Part III. Complexity on the brane

“…where G N is the Newton's constant of bulk gravity theory and is some undetermined length scale. Various aspects of the CV proposal have been studied on the gravitational side of the duality, e.g., see [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. The above conjecture assumes that the state in question is a pure state defined on a global time slice, i.e., the time slice S spans the entire asymptotic boundary.…”

“…Both the holographic CV proposal (1.1) and our proposed generalization (1.6) involve an undetermined length scale . In most previous studies, e.g., [9][10][11][12][13][14][15][16][17][18][19][20][21][22], 19 this length scale is simply chosen to be the AdS curvature scale. However, our analysis was simplified by leaving undetermined, and in particular, we found a simple relation (2.31) between the scales associated with the holographic complexity in the bulk and on the brane.…”

“…where H matter ζ is the matter Hamiltonian associated with the flow generated by the conformal Killing vector ζ on the causal diamond, A is the area of the edge ∂Σ, and k denotes the extrinsic curvature of ∂Σ embedded in the maximal slice. Connections to the first law of holographic complexity were also developed in [17,19,157]. Furthermore, it was extended to higher-curvature gravity in [125] as…”

Semi-classical thermodynamics of quantum extremal surfaces in Jackiw-Teitelboim gravity

“…Recall the CFT vacuum state reduced to a ball on flat space is dual to AdS-Rindler space, such that the first law of entanglement is dual to the first law of AdS-Rindler space [67,68]. AdS-Rindler space is identical to the massless static hyperbolic AdS black hole, so the Smarr formula (4.18) reduces to (D − 2)κA = 2ΘΛ and the extended first law (4.53) can be written as [127,128]…”

“…As proven in [67] and explained below, the small and large AdS-Rindler wedges can be transformed into each other by an isometry, corresponding to a boost in the (T 1 , X) plane in embedding space. Below we mainly follow the derivation of the boost Killing vector in appendix C.2 of [128], but we use different coordinates and we extend the analysis to nested Rindler wedges centered at arbitrary time slices.…”