Entanglement Entropy in Loop Quantum Gravity
William Donnelly
TL;DR
The paper defines entanglement entropy for the gravitational field in loop quantum gravity using spin-network states and the Schmidt decomposition, showing that across a region $\Omega$ the entropy is finite and scales with the number of boundary punctures via $S_E(\Omega) = \sum_{p=1}^P \log(2 \tilde{\jmath}_p + 1)$. It derives the corresponding Schmidt rank $N = \prod_{p=1}^P (2 \tilde{\jmath}_p + 1)$ and analyzes the case where boundary intertwiners contribute additively as $S_E(\Omega) = \sum_{p=1}^P S_E(i_p)$, reducing to the puncture formula when intertwiners are identity-like. The work reveals an asymptotic agreement with the isolated horizon framework for horizon entropy, while identifying a mismatch arising from the spin-projection constraint on boundary intertwiners. It also proposes a method to study corrections to the area law and explores potential links to quantum corrections of the gravitational action, possibly via a geometric quantity akin to a Noether charge.
Abstract
The entanglement entropy between quantum fields inside and outside a black hole horizon is a promising candidate for the microscopic origin of black hole entropy. We show that the entanglement entropy may be defined in loop quantum gravity, and compute its value for spin network states. The entanglement entropy for an arbitrary region of space is expressed as a sum over punctures where the spin network intersects the region's boundary. Our result agrees asymptotically with results previously obtained from the isolated horizon framework, and we give a justification for this agreement. We conclude by proposing a new method for studying corrections to the area law and its implications for quantum corrections to the gravitational action.