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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 ideathat the area of event horizons might be related to an entropy. Here the eventhorizon 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 factthat CHI is bounded below by the von Neumann entropy $S$ suggests that CHI iscoarse-grained. Its properties could thus differ markedly from those of $S$. Inparticular, recent results imply that when $d\le 4$ holographic QFTs areperturbatively coupled to $d$-dimensional gravity, the combined systemsatisfies the so-called quantum focusing condition (QFC) at leading order inthe new gravitational coupling $G_d$ when the QFT entropy is taken to be thatof von Neumann. However, by studying states dual to spherical bulk (anti--deSitter) Schwarschild black holes in the conformal frame for which the boundaryis a $(2+1)$-dimensional de Sitter space, we find the QFC defined by CHI isviolated even when perturbing about a Killing horizon and using a single nullcongruence. Since it is known that a generalized second law (GSL) holds in thiscontext, our work demonstrates that the QFC is not required in order for anentropy, or an entropy-like quantity, to satisfy such a GSL.

Causal holographic information does not satisfy the linearized quantum focusing condition