Information Graph Flow: a geometric approximation of quantum and statistical systems

TL;DR

This paper introduces a method to approximate complex quantum and statistical systems with low-dimensional geometric field theories by constructing and deforming information graphs based on mutual information, revealing emergent geometry.

Contribution

It proposes a novel three-step procedure to derive geometric degrees of freedom from large systems using information graphs and graph flow equations, with potential applications in quantum gravity.

Findings

Demonstrates geometric attractors towards 1D and 2D lattices

Shows how graph flow equations produce sparse adjacency matrices

Suggests a new approach to emergent quantum gravity

Abstract

Given a quantum (or statistical) system with a very large number of degrees of freedom and a preferred tensor product factorization of the Hilbert space (or of a space of distributions) we describe how it can be approximated with a very low-dimensional field theory with geometric degrees of freedom. The geometric approximation procedure consists of three steps. The first step is to construct weighted graphs (we call information graphs) with vertices representing subsystems (e.g. qubits or random variables) and edges representing mutual information (or the flow of information) between subsystems. The second step is to deform the adjacency matrices of the information graphs to that of a (locally) low-dimensional lattice using the graph flow equations introduced in the paper. (Note that the graph flow produces very sparse adjacency matrices and thus might also be used, for example, in machine learning or network science where the task of graph sparsification is of a central importance.) The third step is to define an emergent metric and to derive an effective description of the metric and possibly other degrees of freedom. To illustrate the procedure we analyze (numerically and analytically) two information graph flows with geometric attractors (towards locally one- and two-dimensional lattices) and metric perturbations obeying a geometric flow equation. Our analysis also suggests a possible approach to (a non-perturbative) quantum gravity in which the geometry (a secondary object) emerges directly from a quantum state (a primary object) due to the flow of the information graphs.