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arXiv · 2601.19857

Symmetric and Antisymmetric Quantum States from Graph Structure and Orientation

Matheus R. de Jesus·Eduardo O. C. Hoefel·Renato M. Angelo
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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 generate fully antisymmetric multipartite states when endowed with appropriate orientations. Together, these results provide a unified graph-theoretic description of bosonic and fermionic exchange symmetry based on graph completeness and edge orientation.

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Matheus R. de Jesus, Eduardo O. C. Hoefel, Renato M. Angelo. 2026-01-27. Symmetric and Antisymmetric Quantum States from Graph Structure and Orientation. https://doi.org/10.3390/e28040386

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