Moufang Patterns and Geometry of Information

TL;DR

This paper explores the symmetries in information geometry, revealing Moufang loop structures in probability spaces, and introduces new algebraic constructions for quantum error correction and tensor networks.

Contribution

It demonstrates that spaces of probability distributions have Moufang loop symmetries and introduces novel algebraic models for quantum codes and tensor networks.

Findings

Probability spaces possess Moufang loop symmetries.

New algebraic constructions for quantum error-correcting codes.

Connection between differential geometry and algebraic structures in information theory.

Abstract

Technology of data collection and information transmission is based on various mathematical models of encoding. The words "Geometry of information" refer to such models, whereas the words "Moufang patterns" refer to various sophisticated symmetries appearing naturally in such models. In this paper we show that the symmetries of spaces of probability distributions, endowed with their canonical Riemannian metric of information geometry, have the structure of a commutative Moufang loop. We also show that the F-manifold structure on the space of probability distribution can be described in terms of differential 3-webs and Malcev algebras. We then present a new construction of (noncommutative) Moufang loops associated to almost-symplectic structures over finite fields, and use then to construct a new class of code loops with associated quantum error-correcting codes and networks of perfect tensors.