Multicategorical Adjoints, Monadicity, and Quantum Resources

TL;DR

This paper develops a unified algebraic framework for quantum processes using multicategorical structures, introducing n-adjoints, a monadic description of quantum maps, and operadic encoding of quantum no-go principles.

Contribution

It introduces n-adjoints for multicategorical operator interactions, proves a monadic characterization of quantum processes, and encodes quantum constraints as operadic ideals, advancing the algebraic foundation of quantum theory.

Findings

Multicategorical n-adjoints extend classical adjoints.

Quantum processes are characterized as algebras of a synergy monad.

Quantum no-go principles are encoded as operadic ideals.

Abstract

This paper is the third part of a program aimed at building a unified operadic and multicategorical foundation for operator theory and quantum processes. Building on the multicategory HilbMult and the previously introduced Synergy Operad, we investigate the intrinsic algebraic structure underlying compositional quantum dynamics. Three main themes are developed. First, we introduce n-adjoints, a hierarchy of generalized adjoint operations that extend classical adjoints to multicategorical settings and capture higher-order operator interactions. We prove a multicategorical version of Stinespring's theorem formulated in terms of n-adjoints. Second, we study the algebraic semantics of quantum processes and show that completely positive trace-preserving maps admit a canonical monadic description. In particular, we prove a monadicity theorem identifying quantum processes, up to operadic operational equivalence, with algebras of a synergy monad induced by a symmetric quantum interaction operad, yielding a faithful representation that uniformly accounts for sequential and parallel composition. Third, we present an algebraic formulation of quantum no-go principles by encoding constraints such as the no-cloning theorem as operadic ideals. Together, these results provide a coherent algebraic framework that unifies operator theory, category theory, and quantum information.