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 (QNEC) 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 (QFC), and gravitational equations of motion. We show that the leading-order gravitational equations of motion, Einstein's equations, are equivalent to the leading-order saturation of the QFC for Planck-width deformations.
Highlights
The connection between quantum information and energy has been an emerging theme of recent progress in quantum field theory
II we review some of the basic concepts of entropy, relative entropy, and the holographic setup that will be relevant for our calculation
We will see that the large-k response of U [and UðdÞ] is completely determined by near-boundary physics and in particular will match the results we found in previous sections
Summary
The connection between quantum information and energy has been an emerging theme of recent progress in quantum field theory. For the special case of an interacting conformal field theory (CFT) this fully specifies the stress tensor in terms of entropy variations: by considering (1.2) for all entangling surfaces passing through a point, hTμνi is completely determined up to a trace term. For a region bounded by an entangling surface restricted to a null plane the modular Hamiltonian has a known formula in terms of the stress tensor [13,14], and in particular we have. It is enough to restrict our attention theories where all relevant couplings have mass dimension greater than d=2, and to states where operators of dimension Δ ≤ d=2 have vanishing expectation values near the entangling surface The idea of these restrictions is to make sure there are no parameters with a scaling dimension small enough to contribute to divergences.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
