Time Evolution of Entanglement Entropy from Black Hole Interiors (1303.1080v2)

Abstract: We compute the time-dependent entanglement entropy of a CFT which starts in relatively simple initial states. The initial states are the thermofield double for thermal states, dual to eternal black holes, and a particular pure state, dual to a black hole formed by gravitational collapse. The entanglement entropy grows linearly in time. This linear growth is directly related to the growth of the black hole interior measured along "nice" spatial slices. These nice slices probe the spacelike direction in the interior, at a fixed special value of the interior time. In the case of a two-dimensional CFT, we match the bulk and boundary computations of the entanglement entropy. We briefly discuss the long time behavior of various correlators, computed via classical geodesics or surfaces, and point out that their exponential decay comes about for similar reasons. We also present the time evolution of the wavefunction in the tensor network description.

• The paper demonstrates that linear growth in entanglement entropy is driven by the expansion of 'nice' spatial slices inside black holes.
• Using methods like the replica trick and tensor networks, the study aligns bulk gravity computations with 2D CFT predictions (v = 1).
• It highlights key implications for quantum thermalization and holographic frameworks, paving the way for deeper quantum gravity insights.

Time Evolution of Entanglement Entropy from Black Hole Interiors

The paper by Hartman and Maldacena ventures into the intricate subject of entanglement entropy's time evolution, particularly associated with black hole interiors, using the conceptual framework provided by the AdS/CFT correspondence. This research elucidates how entangled quantum states transmute over time, drawing substantive insights from gravitational dynamics within black hole interiors.

The paper primarily focuses on two black hole configurations: the eternal black hole and a single-sided black hole formed by gravitational collapse. These correspond to thermal and pure states in the boundary Conformal Field Theory (CFT), respectively. The core finding is the linear growth of entanglement entropy over time, which the authors connect to the expansion of "nice" spatial slices traversing the black hole interior. These nice slices extend in the spacelike direction, contributing to the linear increase in entropy.

In the context of a two-dimensional CFT, Hartman and Maldacena demonstrate alignment between computations in the bulk (gravity side) and boundary (CFT side), specifically noting that the speed of entanglement propagation in these setups equals the speed of light (v = 1). This exact match underscores the consistency of their theoretical modeling with established CFT results, particularly referencing the Cardy-Calabrese calculations.

The paper also explores the eventual saturation of entanglement at a thermal value, occurring at a time proportional to the size of the spatial region considered. This saturation reflects a transition from short-range entanglement in the initial states to widespread entanglement over larger scales as the system evolves.

The methodological rigor is evident in the treatment of various computational techniques such as the replica method and tensor network descriptions. On the gravity side, the examination involves extremal surfaces in Anti-de Sitter (AdS) space, extending previously established rules for static spacetimes to dynamic settings, and applying them to thermalization processes in strongly coupled CFTs.

The implications of this research proliferate across theoretical and practical domains. The linear growth of entanglement entropy could provide insights into quantum thermalization and serve as a metric for information flow within quantum gravity systems. The results also offer an avenue to explore tensor network descriptions for CFTs, inspired by the behavior of entanglement entropy. These networks propose a layered structure that grows with time, analogous to the expansion of nice slices within the black hole.

The investigation into black hole interiors and entanglement touches on fundamental aspects of holography and quantum information theory, providing potential paths forward for understanding spacetime's quantum nature. Future research might explore nonspherical configurations or extend these findings to higher-dimensional theories, thus broadening the scope and applicability of these pivotal conclusions.