Abstract
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.
Highlights
Motivated by the increasing connections between information theory and high-energy physics, in the context of the AdS/CFT correspondence, we explore the information geometry associated to a variety of simple systems
The probability distribution is replaced with the density matrix of the quantum states whose symmetries take the form of conjugation by an appropriate unitary matrix. These two issues – symmetry and non-uniqueness – immediately raise two questions: first, how much physics of the field theory is encoded in the information geometry? Second, can we find other meaningful realizations of the information space associated to a given theory, and perhaps even generalize gauge/gravity duality to an “information/geometry” duality applicable to a wider class of theories? The purpose of this work is to present some initial explorations into these questions, as well as to collect some facts about the Fisher metric which, while familiar to experts in information geometry, do not appear to have survived the latter’s recruitment into the quantum field theory community
When we say that the information geometry merely reflects the underlying symmetries of the distribution, we mean that a symmetry of the probability distribution will manifest itself as a corresponding symmetry of the Fisher metric
Summary
Recent progress in understanding the AdS/CFT correspondence has seen an explosion of effort at the interface of information theory and both quantum field theory and gravity. We do this for the quantum analogue of the Fisher metric on a space of quantum states. [42] asserts that the geometry of non-interacting models is flat, while this is clearly false for even the simple Gaussian example mentioned above We believe this is a confusion of language, stemming from the fact that in information geometry, one typically considers the 1-connection, rather than the 0-connection familiar to physicists; and the associated 1-curvature is zero for a wide class of models, known as exponential families (see sections 2.1).
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