Calibrated hypergraph states: I calibrated hypergraph and multi qudit state monads
Roberto Zucchini
TL;DR
The paper develops a rigorous, category-theoretic foundation for calibrated hypergraph states and multi-qudit states by introducing graded $\varOmega$ monads that pair hypergraph combinatorics with quantum-state formalisms. It defines three hypergraph monads $G\varOmega$, $G_C\varOmega$, and $G_W\varOmega$ to encode bare hypergraphs, calibrated hypergraphs, and weighted hypergraphs within a unified graded-monad setting, and then builds the associated multi-dit configuration monad $E\varOmega$ and multi-qudit state monad ${{\fam=12 H}}_E\varOmega$ to capture classical configurations and quantum states. A key contribution is showing that calibrated hypergraphs and multi-qudit states organize as graded $\varOmega$ monads, with a calibration-to-weight linkage $\sfh$ and natural projections to the base $G\varOmega$, thereby enabling a canonical calibrated hypergraph state map to be constructed in Part II. This framework provides a constructive, abstract foundation for analyzing hypergraph-based quantum states, potentially enabling new insights into entanglement structures and generalizations to higher qudits and Galois rings. The approach foregrounds a deep connection between Pro/finite-ordinal category theory and quantum information constructs, offering a principled path toward systematic state map constructions and entanglement classifications.
Abstract
Hypergraph states are a special kind of multipartite states encoded by hypergraphs. They play a significant role in quantum error correction, measurement--based quantum computation, quantum non locality and entanglement. In a series of two papers, we introduce and study calibrated hypergraph states, a broad generalization of weighted hypergraph states codified by hypergraphs equipped with calibrations, an ample extension of weightings. We propose as a guiding principle that a constructive theory of hypergraph states must be based on a categorical framework for hypergraphs on one hand and multi qudit states on the other constraining hypergraph states enough to render the determination of their general structure possible. In this first paper, we introduce graded $\varOmega$ monads, concrete Pro categories isomorphic to the Pro category $\varOmega$ of finite von Neumann ordinals and equipped with an associative and unital graded multiplication, and their morphisms, maps of $\varOmega$ monads compatible with their monadic structure. We then show that both calibrated hypergraphs and multi qudit states naturally organize in graded $\varOmega$ monads. In this way, we lay the foundation for the construction of calibrated hypergraph state map as a special morphism of these $\varOmega$ monads in the companion paper.