Bekenstein Bound for Approximately Local Charged States

Abstract

We generalize the energy-entropy ratio inequality in quantum field theory (QFT) established by one of us from localized states to a larger class of states. The states considered in this paper can be in a charged (non-vacuum) representation of the QFT or may be only approximately localized in the region under consideration. Our inequality is $S(\hat{I}¨||\hat{I}\text{c◯})\le 2\ddot{I}€R(\hat{I}¨,H_{\ddot{I}}\hat{I}¨)+\log d(\ddot{I})+\epsilon$, where $S$ is the relative entropy, where $R$ is a "radius" (width) characterizing the size of the region, $d(\ddot{I})$ is the statistical (quantum) dimension of the given charged sector $\ddot{I}$ hosting the quantum state $\hat{I}¨$, $\hat{I}\text{c◯}$ is the vacuum state, $H_{\ddot{I}}$ is the Hamiltonian in the charged sector, and $\epsilon$ is a tolerance measuring the deviation of $\hat{I}¨$ from the vacuum according to observers in the causal complement of the region.

24 pages, no figures