Graphical description of unitary transformations on hypergraph states

TL;DR

This paper develops a graphical framework for understanding how nonlocal unitary transformations act on hypergraph states, generalizing existing graph state techniques and revealing new insights into their equivalence and entanglement properties.

Contribution

It introduces graphical rules for nonlocal unitary transformations on hypergraph states, extending local complementation concepts and enabling new methods for entanglement detection.

Findings

Local Pauli operations are insufficient to determine hypergraph state equivalence for five qubits.

Graphical rules for gates like CNOT and Toffoli are established.

Entanglement witnesses for three-uniform hypergraph states are constructed.

Abstract

Hypergraph states form a family of multiparticle quantum states that generalizes cluster states and graph states. We study the action and graphical representation of nonlocal unitary transformations between hypergraph states. This leads to a generalization of local complementation and graphical rules for various gates, such as the CNOT gate and the Toffoli gate. As an application, we show that already for five qubits local Pauli operations are not sufficient to check local equivalence of hypergraph states. Furthermore, we use our rules to construct entanglement witnesses for three-uniform hypergraph states.