Symmetric and Antisymmetric Quantum States from Graph Structure and Orientation

TL;DR

This paper establishes a graph-theoretic framework linking graph structure and orientation to symmetric and antisymmetric quantum states, unifying descriptions of bosonic and fermionic exchange symmetry.

Contribution

It introduces a novel construction using directed graphs and a non-commutative gate to generate antisymmetric states, extending the standard symmetric graph state framework.

Findings

Complete graphs correspond to symmetric states.

Directed graphs with specific orientations generate antisymmetric states.

Unified graph-theoretic description of bosonic and fermionic symmetry.

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.