Symmetric and Antisymmetric Quantum States from Graph Structure and Orientation

Area of Science:

• Quantum Information Science
• Quantum Computing
• Quantum Entanglement

Background:

• Graph states are fundamental for multipartite entanglement in quantum information.
• Standard graph states, generated via controlled-Z interactions, exhibit symmetric exchange properties.
• A direct link between graph topology and exchange symmetry was previously lacking.

Purpose of the Study:

• To establish a precise correspondence between graph topology and exchange symmetry in quantum states.
• To introduce a generalized graph-based construction for multipartite states.
• To provide a unified graph-theoretic description of bosonic and fermionic exchange symmetry.

Main Methods:

• Proving that graph states are fully symmetric if and only if the underlying graph is complete.
• Introducing a generalized graph-based construction using a non-commutative two-qudit gate (GR).
• Utilizing directed edges and explicit vertex ordering in the generalized construction.

Main Results:

• A complete graph corresponds to a fully symmetric graph state.
• Complete directed graphs with appropriate orientations generate fully antisymmetric multipartite states.
• The study provides a unified graph-theoretic framework for both symmetric (bosonic) and antisymmetric (fermionic) exchange properties.

Conclusions:

• Graph topology precisely dictates the exchange symmetry of quantum states.
• The generalized construction unifies the description of bosonic and fermionic exchange symmetry.
• This work offers a powerful graph-theoretic tool for understanding and generating entangled quantum states.