  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.
