Non-symmetric GHZ states: weighted hypergraph and controlled-unitary graph representations

Non-symmetric GHZ states ($n$-GHZ$_\alpha$), defined by unequal superpositions of $|00...0>$ and $|11...1>$, 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 non-symmetric 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 non-symmetric GHZ states can be efficiently stabilized using local operations and a single ancilla, independent of system size. We extend this framework to qudit systems, constructing LU-equivalent weighted qudit hypergraphs and showing that general non-symmetric qudit GHZ states can be described as star-shaped CU graphs. Our results provide a systematic approach to characterizing and stabilizing non-symmetric multipartite entanglement in both qubit and qudit systems, with implications for quantum error correction and networked quantum protocols.