Genuine entanglement and LU-equivalence of graph states in bipartition

Graph states are intuitive representations of quantum states that are useful in quantum computing and quantum information. In this paper, we investigate the role of entanglement and local unitary equivalence in quantum computing through graph states in high-dimensional Hilbert spaces. We introduce bipartitions on the vertices sets and demonstrate separability and local unitary equivalence based on different choices of partitions. By performing the two-qubit controlled phase shift gates on various pairs of multiqubit system, one can generate a genuinely entangled graph state based on simple connected graphs. We extend 2-dimensional Hilbert spaces to higher and mixed dimensions and examine pure quantum states that cannot be separated by any bipartition. In addition, we generalize the analysis on classification of local unitary equivalence beyond the controlled-Z gates to controlled phase gates. By examining graphs constructed from bipartition graphs, we have shown that graphs with identical bipartition subgraph structures are local unitary equivalent.