Symmetric and Antisymmetric Quantum States from Graph Structure and Orientation

Abstract: Graph states provide a powerful framework for describing multipartite entanglement in quantum information science. In their standard formulation, graph states are generated by controlled-$Z$ interactions and naturally encode symmetric exchange properties. Here we establish a precise correspondence between graph topology and exchange symmetry by proving that a graph state is fully symmetric under particle permutations if and only if the underlying graph is complete. We then introduce a generalized graph-based construction using a non-commutative two-qudit gate, denoted $GR$, which requires directed edges and an explicit vertex ordering. We show that complete directed graphs endowed with appropriate orientations, for an odd number of qudits generate fully antisymmetric multipartite states. Together, these results provide a unified graph-theoretic description of bosonic and fermionic exchange symmetry based on graph completeness and edge orientation.