The Gravity Dual of a Density Matrix

TL;DR

The paper investigates how much of a bulk spacetime in AdS/CFT can be reconstructed from the density matrix rho_A of a boundary region A. It formalizes R(A) as the largest bulk region determined by rho_A and derives general constraints, showing that the causal wedge z(D_A) (and its domain) must be contained in R(A) while entanglement-based constructs like the wedge of extremal surfaces w(D_A) can extend reconstruction beyond causal bounds. The authors propose hat{w}(D_A) as a plausible maximal region under general conditions, but also present cases where R(A) must exceed hat{w}(D_A), illustrating intricate ties between entanglement structure and bulk geometry. Overall, the work highlights entanglement as essential to spacetime emergence and outlines a framework for understanding bulk reconstruction from boundary data beyond naive locality.

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.

Figures (8)

• Figure 1: A spacelike slice $\Sigma$ of a boundary manifold B ($= S^1 \times$ time) with a region $A$ and its domain of dependence $D_A$. The same domain of dependence arises from any spacelike boundary region $\tilde{A}$ homologous to $A$ with $\partial A = \partial \tilde{A}$.
• Figure 2: Causal wedge $z(D_A)$ associated with a domain of dependence $D_A$.
• Figure 3: In pure global AdS, causal wedges of complementary hemispherical regions of the $\tau=0$ slice intersect along a codimension-two surface. In generic asymptotically AdS spacetimes, they intersect only at the boundary.
• Figure 4: Different possible behaviors of extremal surfaces in spherically symmetric static spacetimes. Shaded region indicates $w(D_A)$ where $A$ is the right hemisphere. The boundary of the shaded region on the interior of the spacetime is the minimal area extremal surface bounded by the equatorial $S^{d-1}$.
• Figure 5: Region $w(D_A)$ (shaded) where $A$ is a boundary sphere of angular size greater than $\pi$. No minimal surface with boundary in $A$ penetrates the unshaded middle region.