Causal holographic information does not satisfy the linearized quantum focusing condition

TL;DR

This paper shows that the causal holographic information (CHI), a quantity related to entropy in holographic theories, does not satisfy the linearized quantum focusing condition (QFC), indicating limitations of CHI as an entropy measure.

Contribution

The study demonstrates that CHI violates the quantum focusing condition in certain holographic setups, challenging its interpretation as a coarse-grained entropy.

Findings

CHI violates the quantum focusing condition in specific holographic states.

The generalized second law holds despite CHI's violation of QFC.

CHI's properties differ from von Neumann entropy in holographic contexts.

Abstract

The Hubeny-Rangamani causal holographic information (CHI) defined by a region $R$ of a holographic quantum field theory (QFT) is a modern version of the idea that the area of event horizons might be related to an entropy. Here the event horizon lives in a dual gravitational bulk theory with Newton's constant $G_{\rm bulk}$, and the relation involves a factor of $4G_{\rm bulk}$. The fact that CHI is bounded below by the von Neumann entropy $S$ suggests that CHI is coarse-grained. Its properties could thus differ markedly from those of $S$. In particular, recent results imply that when $d\le 4$ holographic QFTs are perturbatively coupled to $d$-dimensional gravity, the combined system satisfies the so-called quantum focusing condition (QFC) at leading order in the new gravitational coupling $G_d$ when the QFT entropy is taken to be that of von Neumann. However, by studying states dual to spherical bulk (anti--de Sitter) Schwarschild black holes in the conformal frame for which the boundary is a $(2+1)$-dimensional de Sitter space, we find the QFC defined by CHI is violated even when perturbing about a Killing horizon and using a single null congruence. Since it is known that a generalized second law (GSL) holds in this context, our work demonstrates that the QFC is not required in order for an entropy, or an entropy-like quantity, to satisfy such a GSL.