Relative entropy equals bulk relative entropy (1512.06431v1)

Abstract: We consider the gravity dual of the modular Hamiltonian associated to a general subregion of a boundary theory. We use it to argue that the relative entropy of nearby states is given by the relative entropy in the bulk, to leading order in the bulk gravitational coupling. We also argue that the boundary modular flow is dual to the bulk modular flow in the entanglement wedge, with implications for entanglement wedge reconstruction.

• The paper establishes that boundary relative entropy equals bulk relative entropy to leading order by analyzing modular Hamiltonians and entanglement wedges.
• It employs the holographic principle and the Ryu-Takayanagi proposal to bridge entropy computations between quantum field theories and their gravitational duals.
• The findings simplify complex entropic calculations and bolster the AdS/CFT correspondence, paving the way for advanced studies in quantum gravity.

An Analysis of the Equivalence Between Boundary and Bulk Relative Entropy

The paper, authored by Jafferis, Lewkowycz, Maldacena, and Suh, presents an insightful exploration of the relationship between boundary and bulk relative entropy within the framework of holographic duality. By exploring the intricacies of entanglement in quantum field theories with gravity duals, the authors put forth a compelling argument for the equivalence of relative entropy on the boundary and in the bulk, to leading order in the bulk gravitational coupling. This paper's critical results lay the groundwork for advancing our understanding of the AdS/CFT correspondence, particularly in the context of entanglement.

Overview of Key Concepts and Results

The central theme of the paper is the investigation of modular Hamiltonians and their gravitational duals. The authors focus on the relative entropy, which quantifies the distinguishability between two states in quantum mechanics. They start by considering the gravity dual of the modular Hamiltonian for a subregion of the boundary theory and assert that this establishes a direct connection to the modular flow within the entanglement wedge—a region of space that plays a significant role in entanglement reconstruction.

The relative entropy, denoted $$S(\rho|\sigma) = \text{Tr}\left[ \rho \log \rho - \rho \log \sigma \right]$$, where $\rho$ and $\sigma$ are density matrices, emerges as a centerpiece. The paper innovatively argues that the relative entropy of states on the boundary matches the relative entropy of their corresponding bulk states, hence $$S_{\rm bdy}(\rho|\sigma) = S_{\rm bulk}(\rho|\sigma)$$, to leading order in the gravitational coupling.

In establishing the modular Hamiltonian relationship, the authors extend prior work to cases beyond symmetries such as thermal states or Rindler space. They claim a generalization that holds for arbitrary regions, with or without a symmetry, and independently of whether the reference state $\sigma$ is the vacuum. This broadened scope aligns with ensuring the equation’s validity to all orders within bulk perturbation theory.

Notable Methodological Approaches

The crux of the argument rests on the holographic principle, particularly the Ryu-Takayanagi proposal and its quantum extensions. At leading order, entanglement entropy is expressed via the area of a minimal surface in the bulk. Their modular Hamiltonian formula incorporates multiple components: the boundary modular Hamiltonian is related to the area of an extremal surface (the Ryu-Takayanagi surface), and the bulk modular Hamiltonian within the entanglement wedge. This leads to the conclusion that, to leading order, the modular Hamiltonian on the boundary simplifies significantly in the bulk context.

When dealing with quantum corrections, the authors integrate additional semiclassical bulk terms, represented as local operators on the extremal surface. Crucially, these contributions cancel out in the relative entropy computation, reinforcing the notion of equivalence between bulk and boundary formulations.

Implications and Future Directions

The implications of these findings are manifold. Practically, they affect how boundary observable entropy calculations can be performed through their bulk equivalents, offering a calculationally feasible path in complex quantum systems. Theoretically, this equivalence reinforces the fidelity of the holographic principle and deepens our understanding of concepts like entanglement wedge reconstruction and quantum error correction in holography.

Looking forward, these efforts offer intriguing speculation about the role of gravitational theories with higher derivatives and extensions to non-extremal surfaces, promising fertile ground for future inquiry. Additionally, applying their findings to scenarios free of symmetries, particularly involving coherent states, presents an equally interesting challenge.

In conclusion, this paper provides a solid foundation in the ongoing endeavor to understand the entropic properties within the holographic paradigm. It simplifies and unifies disparate aspects of boundary-bulk relationships through the lens of relative entropy, contributing a pivotal piece to the broader puzzle of quantum gravity.