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arXiv:2106.11313

Infinite Distance Limits and Information Theory

John Stout

Published Jun 21, 2021• 11 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2106.11313

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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.
We study the question of how reliably one can distinguish two quantum field theories (QFTs). Each QFT defines a probability distribution on the space of fields. The relative entropy provides a notion of proximity between these distributions and quantifies the number of measurements required to distinguish between them. In the case of nearby conformal field theories, this reduces to the Zamolodchikov metric on the space of couplings. Our formulation quantifies the information lost under renormalization group flow from the UV to the IR and leads us to a quantification of fine-tuning. This formalism also leads us to a criterion for distinguishability of low energy effective field theories generated by the string theory landscape.
We study the question of how reliably one can distinguish two quantum field theories (QFTs). Each QFT defines a probability distribution on the space of fields. The relative entropy provides a notion of proximity between these distributions and quantifies the number of measurements required to distinguish between them. In the case of nearby conformal field theories, this reduces to the Zamolodchikov metric on the space of couplings. Our formulation quantifies the information lost under renormalization group flow from the UV to the IR and leads us to a quantification of fine-tuning. This formalism also leads us to a criterion for distinguishability of low energy effective field theories generated by the string theory landscape.
We study the question of how reliably one can distinguish two quantum field theories (QFTs). Each QFT defines a probability distribution on the space of fields. The relative entropy provides a notion of proximity between these distributions and quantifies the number of measurements required to distinguish between them. In the case of nearby conformal field theories, this reduces to the Zamolodchikov metric on the space of couplings. Our formulation quantifies the information lost under renormalization group flow from the UV to the IR and leads us to a quantification of fine-tuning. This formalism also leads us to a criterion for distinguishability of low energy effective field theories generated by the string theory landscape.
Niels Bohr supposedly said that if quantum mechanics did not make you dizzy then you did not understand it. I think that the same can be said about statistical inference. Robert D. Cousins In this chapter we quantify how easy it may be too distinguish probability distributions from each other (a discussion that was started in Chapter 2). The issue is a very practical one and arises whenever one is called upon to make a decision based on imperfect data. There is no unique answer because everything depends on the data – the l 1 -distance appears if there has been just one sampling of the distributions, the relative entropy governs the approach to the ‘true’ distribution as the number of samplings goes to infinity, and so on. The quantum case is even subtler. A quantum state always stands ready to produce a large variety of classical probability distributions, depending on the choice of measurement procedure. It is no longer possible to distinguish pure states from each other with certainty, unless they are orthogonal. The basic idea behind the quantum distinguishability measures is the same as that which allowed us, in Section 5.3, to relate the Fubini–Study metric to the Fisher–Rao metric. We will optimize over all possible measurements. Classical distinguishability measures If a distance function has an operational significance as a measure of statistical distinguishability, then we expect it to be monotone (and decreasing) under general stochastic maps. Coarse graining means that information is being discarded, and this cannot increase distinguishability.
Quantum field theory assigns infinite-dimensional Hilbert spaces to finite spatial regions, while gravitational physics implies strict bounds on physically distinguishable states. This work introduces configuration capacity, a resolution-dependent operational measure of distinguishability defined using completely positive, trace-preserving coarse-graining channels and the trace distance. Configuration capacity quantifies the maximum number of operationally distinguishable configurations accessible to a physical system at finite resolution. It provides a conservative framework for reconciling infinite Hilbert-space structure with finite physical information without introducing new degrees of freedom or speculative microscopic assumptions. The concept is illustrated explicitly for harmonic oscillator systems, where finite experimental resolution enforces finiteness despite infinite Hilbert-space dimension. Standard entropy measures—including thermodynamic entropy, entanglement entropy, and Bekenstein–Hawking entropy—emerge as limiting cases or upper bounds under appropriate coarse-grainings. The framework further clarifies that gravitational interaction modifies unitary evolution through spacetime geometry but does not constitute a measurement, and that gravitational entropy bounds can be interpreted as ceilings on operational distinguishability rather than microscopic state multiplicity. This manuscript is intended for journal submission; the present version serves as an openly accessible preprint.
The exchange of information between an open quantum system and its environment, especially the backflow of information from the environment to the open system associated with quantum notions of non-Markovianity, is a widely discussed topic for years now. This information flow can be quantified by means of the trace distance of pairs of quantum states which provides a measure for the distinguishability of the states. The same idea can also be used to characterize the information flow in classical open systems through a suitable distance measure for their probability distributions on phase space. Here, we investigate the connection between the trace distance based quantum measure and the Kolmogorov distance for differently ordered quasiprobability distributions on phase space. In particular, we show that, for any pair of quantum states, one can find a unique quasiprobability distribution for which the Kolmogorov distance coincides with the trace distance. We further study the quantum-to-classical transition of the distance measures. By employing the Caldeira-Leggett model of quantum Brownian motion as a prototypical example, numerical simulations indicate a particularly rapid convergence of the Kolmogorov distance of the Wigner functions to the trace distance in the classical uncertainty limit, which establishes the Wigner function distance as an optimal tool for measuring semi-classical information backflow and for quantifying non-Markovianity in open continuous variable quantum systems.