The gravity dual of a density matrix

Abstract

For a state in a quantum field theory on some spacetime, we can associate a density matrix to any subset of a given spacelike slice by tracing out the remaining degrees of freedom. In the context of the AdS/CFT correspondence, if the original state has a dual bulk spacetime with a good classical description, it is natural to ask how much information about the bulk spacetime is carried by the density matrix for such a subset of field theory degrees of freedom. In this note, we provide several constraints on the largest region that can be fully reconstructed, and discuss specific proposals for the geometric construction of this dual region.

Conformal complementarity maps

“…This happens for k = 0 at a single point on S d−1 , whereas the infinite injection of energy occurs in the k = −1 case along the complete equatorial S d−2 which separates S d−1 into two hemispheres. It follows that singularities of 'firewall' type are expected in the global bulk description of such states, in agreement with the philosophy expressed in [42,43]. The price we pay for the ability to use a classical set up is the need to realize the state in a CFT perturbed by a large relevant operator, but the take-away message ends up being the same.…”

Bulk locality and quantum error correction in AdS/CFT

“…In other words we can think of local bulk operators as logical operations on an encoded subspace, which becomes better and better protected against localized boundary errors as we move the operators inwards in the radial direction. 12 We illustrate this in figure 4. In the remainder of the paper we will spell out this idea in more detail, giving the bulk versions of most of the statements of the previous section.…”

“…We will observe that this construction has several paradoxical features, which we will illuminate by recasting it on the CFT side in the language of quantum error correcting codes [9,10]. This language gives a new, more general perspective on the issue of bulk reconstruction, and we believe that it is the natural framework for understanding the idea of "subregion-subregion" duality [11][12][13][14]. In particular, the radial direction in the bulk is realized in the CFT as a measure of how well CFT representations of bulk quantum information are protected from local erasures.…”

“…Our proposal is that we should pick some finite set of local bulk operators φ i (x), realized in the CFT via the 12 One detail here is that in defining the "center" of AdS we are implicitly choosing a conformal frame in the CFT; otherwise AdS is a homogeneous space and no point is special. This is also implicit in our notion of "small" and "large" erasures in the CFT.…”

“…In fact there have been recent conjectures in the literature that this operator can still be reconstructed in A as long as A is bigger than A; more generally, the claim is that one can do reconstruction throughout the entanglement wedge, which is defined as the bulk domain of dependence of any bulk spacelike surface whose boundary is the union of A and the codimension two extremal-area surface of minimal area whose boundary is ∂A [12,[34][35][36]. 16 In the figure, the intersection of the entanglement wedge with a bulk Cauchy surface 16 One argument that this must be the case is as follows: the extension by [37] of the Ryu-Takayanagi proposal [38] to next to leading order in 1/N claims that bulk entanglement entropy in the entanglement wedge contributes to the Von Neumann entropy SA.…”