Symmetric and Antisymmetric Quantum States from Graph Structure and Orientation

TL;DR

The paper investigates how permutation exchange symmetry of multipartite quantum states is encoded by graph topology and orientation. It proves that in the standard CZ-based graph-state formalism, full permutation symmetry occurs if and only if the graph is complete, and that antisymmetric states cannot be realized in this framework. It then introduces a GR gate with directed graphs and explicit vertex ordering, showing that complete directed graphs with appropriate orientation generate fully antisymmetric states for odd numbers of qudits, thereby unifying bosonic and fermionic symmetry in a graph-theoretic language. This generalized construction opens new directions for antisymmetric-state generation, directed-graph MBQC, and potential applications to fermionic networks and symmetry-aware quantum protocols. Overall, the work provides a rigorous link between graph completeness, edge orientation, and exchange symmetry in quantum states, expanding the graph-state paradigm beyond its symmetric origins.

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.

Figures (3)

• Figure 1: Directed graph illustrating the intermediate step of the recursive construction of $\ket{\Gamma_3}$. The state $\ket{\Gamma_2}$ associated with vertices $1$ and $2$ is extended by adding vertex $3$, initially prepared in the state $H_3\ket{0}_3$. The directed edge $3\to1$ represents the application of the gate $GR^{(3,1)}$. At this stage, the resulting three-qudit state does not exhibit any particular exchange symmetry.
• Figure 2: Initial two-vertex directed subgraph $\Gamma_2$ used as the base case of the recursive construction. Vertex $1$ is prepared in the state $\ket{1}$, while vertex $2$ is prepared in the state $\frac{1}{\sqrt{2}}(\ket{1}-\ket{0})=-H(X\ket{0})$. The directed edge from vertex $2$ to vertex $1$ encodes the application of the two-qudit gate $GR^{(2,1)}$, resulting in the antisymmetric state $\ket{\Gamma_2}=\frac{1}{\sqrt{2}}(\ket{01}-\ket{10})$.
• Figure 3: Directed complete graph on three vertices whose orientation generates the fully antisymmetric state of three $d=3$ qudits. The directed edges represent the ordered application of the gates $GR^{(3,1)}$ and $GR^{(3,2)}$, leading to the state $\ket{\Gamma_3}=\ket{A_3}$.