Quantum Operads

TL;DR

This paper introduces quantum symmetries of operads, extending classical symmetry concepts to quantum contexts, with applications to quantum information structures like states and codes.

Contribution

It develops a framework for quantum symmetries of operads and explores their applications in quantum information theory.

Findings

Quantum symmetries of operads are formally defined.

Quantum states and codes can be modeled as operad algebras.

The framework connects categorical structures with quantum information.

Abstract

The most standard description of symmetries of a mathematical structure produces a group. However, when the definition of this structure is motivated by physics, or information theory, etc., the respective symmetry objects might become more sophisticated: quasigroups, loops, quantum groups, ... In this paper, we introduce and study quantum symmetries of very general categorical structures: operads. Its initial motivation were spaces of probability distributions on finite sets. We also investigate here how structures of quantum information, such as quantum states and some constructions of quantum codes are algebras over operads.