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

Non-symmetric GHZ states ($n$-GHZ$_\alpha$), characterized by unequal superpositions of $|00...0>$ and $|11...1>$, represent a significant yet underexplored class of multipartite entangled states with potential applications in quantum information. Despite their importance, the lack of a well-defined stabilizer formalism and corresponding graph representation has hindered their comprehensive study. In this paper, we address this gap by introducing two novel graph formalisms and stabilizers for non-symmetric GHZ states. First, we provide a weighted hypergraph representation and demonstrate that non-symmetric GHZ states are local unitary (LU) equivalent to fully connected weighted hypergraphs. Although these weighted hypergraphs are not stabilizer states, we show that they can be stabilized using local operations, and an ancilla. We further extend this framework to qudits, offering a specific form for non-symmetric qudit GHZ states and their LU equivalent weighted qudit hypergraphs. Second, we propose a graph formalism using controlled-unitary (CU) operations, showing that non-symmetric qudit GHZ states can be described using star-shaped CU graphs. Our findings enhance the understanding of non-symmetric GHZ states and their potential applications in quantum information science.