Information geometry in quantum field theory: lessons from simple examples

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.

Complexity of mixed Gaussian states from Fisher information geometry

“…The complexity of pure states in quantum field theories has been explored in various studies [21,[30][31][32][33][34][35][36][37][38][39][40][41][42][43] and it would be instructive to extend these analyses to mixed states. The tools of Information Geometry, that we have largely employed in our analysis, could provide further tools to handle this interesting problem [163].…”

Disorder Averaging and its UV (Dis)Contents

“…This is related to the broader question of how well a low energy observer can ever hope to reconstruct a given UV completion, see, e.g.,[14,[40][41][42][43][44] as well as[45][46][47][48][49][50][51][52].…”

Weighted Relative Group Entropies and Associated Fisher Metrics

“…(iv) deepening the differential geometric study of the previous new metrics, and finding new geometric invariants for the modeling and the control of statistical phenomena. For example, the study of the distance related topics (see [ 35 ]), of the geodesics (see [ 36 , 37 , 38 ]) and of the various types of curvature tensor fields (see [ 10 , 39 , 40 , 41 , 42 ] for samples of the needed geometric tools).…”