  Motivated by the increasing connections between information theory and
high-energy physics, particularly in the context of the AdS/CFT correspondence,
we explore the information geometry associated to a variety of simple systems.
By studying their Fisher metrics, we derive some general lessons that may have
important implications for the application of information geometry in
holography. We begin by demonstrating that the symmetries of the physical
theory under study play a strong role in the resulting geometry, and that the
appearance of an AdS metric is a relatively general feature. We then
investigate what information the Fisher metric retains about the physics of the
underlying theory by studying the geometry for both the classical 2d Ising
model and the corresponding 1d free fermion theory, and find that the curvature
diverges precisely at the phase transition on both sides. We discuss the
differences that result from placing a metric on the space of theories vs.
states, using the example of coherent free fermion states. We compare the
latter to the metric on the space of coherent free boson states and show that
in both cases the metric is determined by the symmetries of the corresponding
density matrix. We also clarify some misconceptions in the literature
pertaining to different notions of flatness associated to metric and non-metric
connections, with implications for how one interprets the curvature of the
geometry. Our results indicate that in general, caution is needed when
connecting the AdS geometry arising from certain models with the AdS/CFT
correspondence, and seek to provide a useful collection of guidelines for
future progress in this exciting area.
