Nonsymmetric Greenberger-Horne-Zeilinger states: Weighted hypergraph and controlled-unitary graph representations

Nonsymmetric GHZ states ($n\text{\ensuremath{-}}{\mathrm{GHZ}}_{\ensuremath{\alpha}}$), defined by unequal superpositions of $|00...0\ensuremath{\rangle}$ and $|11...1\ensuremath{\rangle}$, naturally emerge in experiments due to decoherence, control errors, and state preparation imperfections. Despite their relevance in quantum communication, relativistic quantum information, and quantum teleportation, these states lack a stabilizer formalism and a graph representation, hindering their theoretical and experimental analysis. We establish a graph-theoretic framework for nonsymmetric GHZ states, proving their local unitary (LU) equivalence to two structures: fully connected weighted hypergraphs with controlled-phase interactions and star-shaped controlled-unitary (CU) graphs. While weighted hypergraphs generally lack stabilizer descriptions, we demonstrate that nonsymmetric GHZ states can be efficiently stabilized using local operations and a single ancilla, independent of system size, thereby enabling entanglement witness construction and offering a pathway toward qudit quantum error correction. We extend this framework to qudit systems, constructing LU-equivalent weighted qudit hypergraphs and showing that general nonsymmetric qudit GHZ states can be described as star-shaped CU graphs. Our results provide a systematic approach to characterizing and stabilizing nonsymmetric multipartite entanglement in both qubit and qudit systems, with implications for quantum error correction and networked quantum protocols.