Calibrated hypergraph states: II calibrated hypergraph state construction and applications

TL;DR

This paper introduces calibrated hypergraph states for qudits over Galois rings, providing a categorical construction and demonstrating their entanglement properties and relation to weighted hypergraph states.

Contribution

It extends hypergraph state theory by constructing calibrated hypergraph states using a categorical framework and analyzing their entanglement and relation to weighted hypergraph states.

Findings

Calibrated hypergraph states are locally maximally entangleable stabilizer states.

They generalize weighted hypergraph states beyond qubits.

Explicit construction over Galois rings demonstrates their unique properties.

Abstract

Hypergraph states are a special kind of multipartite states encoded by hypergraphs relevant in quantum error correction, measurement--based quantum computation, quantum non locality and entanglement. In a series of two papers, we introduce and investigate calibrated hypergraph states, an extension of weighted hypergraph states codified by hypergraphs equipped with calibrations, a broad generalization of weightings. The guiding principle informing our approach is that a constructive theory of hypergraph states must be based on a categorical framework for both hypergraphs and multi qudit states constraining hypergraph states enough to render the determination of their general structure possible. In this second paper, we build upon the graded $\varOmega$ monadic framework worked out in the companion paper, focusing on qudits over a generic Galois ring. We explicitly construct a calibrated hypergraph state map as a special morphism of the calibrated hypergraph and multi qudit state $\varOmega$ monads. We further prove that the calibrated hypergraph states so yielded are locally maximally entangleable stabilizer states, elucidate their relationship to weighted hypergraph states, show that they reduce to the weighted ones in the familiar qubit case and prove through examples that this is no longer the case for higher qudits.