Infinite Distance Limits and Information Theory

Abstract

The classical information metric provides a unique notion of distance on the space of probability distributions with a well-defined operational interpretation: two distributions are far apart if they are readily distinguishable from one another. The quantum information metric generalizes this to the space of quantum states, and thus defines a notion of distance on an arbitrary continuous family of quantum field theories via their vacua that is proportional to the metric on moduli space when restricted appropriately. In this paper, we study this metric and its operational interpretation in a variety of examples. We specifically focus on why and how infinite distance singularities appear. We argue that two theories are infinitely far apart if they are hyper-distinguishable: that is, if they can be distinguished from one another, with certainty, using only a few measurements. We explain why such singularities appear for the simple harmonic oscillator yet are absent for quantum field theories near a typical quantum critical point, and show how an infinite distance point can emerge when a tower of fields degenerates in mass. Finally, we use this perspective to provide a potential bottom-up motivation for the Swampland Distance Conjecture and indicate how we might extend it beyond current lampposts.

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].…”

Emergent strings at infinite distance with broken supersymmetry

“…Taking into account the volume factor in eq. ( 14), this differs by a factor of a −2 from the "intensive" metric of [69], which is thus finite. At this point, the k → 0 asymptotics of the Fourier modes of h is needed in order to evaluate the IR distance.…”

“…Similarly to our analysis of KK masses, one is thus led to seek a suitable generalization of the Zamolodchikov metric, since there is no exact conformal manifold. Such a notion was introduced by O'Connor and Stephens on information-theoretic grounds [64], explored by Dolan [166,167] and subsequently revisited in numerous settings, most recently by Stout in the context of the distance conjecture [69] [168].…”

“…This approach to the geometry of theory space has been extended by O'Connor and Stephens [64] in the context of quantum information theory, and by Anselmi [65][66][67][68] in the context of renormalization group (RG) flows. The former metric has been recently revisited by Stout [69] (see also [70,71]), and in this work we shall employ both metrics to explore infinite-distance limits in the absence of supersymmetry.…”

Misanthropic Entropy and Renormalization as a Communication Channel

“…On general grounds, one might ask to what extent a set of IR observables can serve to constrain the content of some candidate UV completion (see e.g. [5][6][7][8][9][10] as well as [11][12][13][14][15][16][17]). This issue is also of general interest in trying to understand the extent to which information is lost in passing from short to long distance scales.…”