Energy is Entanglement

TL;DR

This paper establishes a deep connection between energy and entropy in theories with gravity duals by showing the saturation of the Quantum Null Energy Condition (QNEC) and its implications for gravitational equations.

Contribution

It derives a formula relating entropy variations to energy conditions, proving saturation of QNEC at leading order and proposing its universal saturation in interacting theories.

Findings

QNEC is saturated in all states at leading order.

Einstein's equations are equivalent to QFC saturation for Planck-width deformations.

The formula holds at leading order in 1/N and is unaffected at higher orders.

Abstract

We compute the local second variation of the von Neumann entropy of a region in theories with a gravity dual. For null variations our formula says that the diagonal part of the Quantum Null Energy Condition is saturated in every state, thus providing an equivalence between energy and entropy. We prove that the formula holds at leading order in 1/N, and further argue that it will not be affected at higher orders. We conjecture that the QNEC is saturated in all interacting theories. We also discuss the special case of free theories, and the implications of our formula for the Averaged Null Energy Condition, Quantum Focusing Conjecture, and gravitational equations of motion. We show that the leading-order gravitational equations of motion, Einstein's equations, are equivalent to leading-order saturation of the QFC for Planck-width deformations.