The Gravity Dual of a Density Matrix (1204.1330v2)

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.

• The paper establishes that boundary density matrices constrain the reconstructable bulk region through the AdS/CFT framework.
• The study compares causal wedge and extremal surface methods to determine which regions of spacetime can be recovered from quantum data.
• The paper highlights the role of entanglement entropy in encoding gravitational dynamics and the emergent properties of spacetime.

The Gravity Dual of a Density Matrix

The paper "The Gravity Dual of a Density Matrix" explores the intriguing intersection of the AdS/CFT correspondence and quantum information theory. It explores how the geometry of a bulk spacetime is encoded in the density matrix of a conformal field theory (CFT) state, specifically looking at what parts of the bulk can be reconstructed from the density matrix corresponding to a subset of degrees of freedom on a spacelike slice of the boundary.

The AdS/CFT correspondence offers a profound framework where a theory of gravity in a (d+1)-dimensional Anti-de Sitter (AdS) space is related to a d-dimensional Conformal Field Theory (CFT) on the boundary of this space. One of the compelling aspects of this duality is the ability to learn about the bulk spacetime geometry from boundary field theory data. The paper addresses a specific question: given a density matrix for a subset of a boundary region in a state $|\Psi \rangle$ with a good classical spacetime $M$, what part of $M$ can be fully reconstructed from this density matrix?

The authors establish several constraints and explore potential ways to associate certain bulk regions with boundary density matrices. Key constraints explored include the requirement that for spatial regions $A$ and $\tilde{A}$ with the same domain of dependence, the corresponding bulk region $R(A)$ should be identical to $R(\tilde{A})$. Furthermore, $R(A)$ should obey causality constraints, ensuring it does not intersect the causal future or past of regions associated with the complement of $A$.

The paper proposes a few candidates for the bulk region represented by a density matrix $\rho_A$:

1. Causal Wedge (z(D_A)): Initially, the authors consider the causal wedge, defined as the intersection of the causal past and future of a domain of dependence $D_A$ on the boundary. This wedge corresponds to a region that a boundary observer could, in principle, access with causal signals and thus reconstruct from boundary data.
2. The Wedge of Extremal Surfaces (w(D_A)): A more expansive possibility is the union of surfaces used to calculate entanglement entropies according to the Ryu-Takayanagi proposal and its covariant form. These surfaces can penetrate deeper into the bulk than the causal wedge suggests, hinting at the necessity of entanglement observables to understand the full bulk region.

The paper examines these proposals in the context of their adherence to the outlined constraints and the physical plausibility of their assumptions. For example, the extremal surfaces provide a broader sense of the bulk reconstruction potential but necessitate a supporting conjecture regarding their spacelike separation properties.

The work further highlights the significant role of entanglement in the framework of spacetime reconstruction, asserting that while density matrices inform us about localized boundary regions, entanglement deeper interconnects these regions, effectively assembling the full geometry of spacetime. The paper suggests that the intricacies of entropy and entanglement directly reflect the geometry of extremal surfaces in the bulk, thus encoding crucial gravitational dynamics.

Overall, "The Gravity Dual of a Density Matrix" contributes to understanding how elements of quantum information, specifically density matrices and their associated entropy, can inform the geometrical properties of a quantum gravity theory. Further developments in this domain could enhance comprehension of the fine structure of spacetime and its emergent properties from boundary quantum states, potentially illuminating new directions in quantum gravity research.