Distinguishing Graph States by the Properties of their Marginals

Graph states are a class of multi-partite entangled quantum states that are ubiquitous in many networking applications; the study of equivalence relations between graph states under local operations aims to provide methods to relate graph states in networked settings. The problem of determining local unitary (LU) equivalence of graph states is in NP, and it remains an open question if efficient general methods are possible. We introduce a family of easy-to-compute LU-invariants based on the marginal structure of the graphs that allow to rule out equivalence of graph states. We show that these invariants can uniquely identify all LU-orbits and entanglement classes of every graph state of 8 qubits or less and discuss how reliable the methods are for more qubit graph states. We also discuss examples of entanglement classes with more nodes, where their marginal structure does not allow us to tell them apart. Additionally, we generalise tools to test local clifford (LC) equivalence of graph states that work by condensing graphs into other graphs of smaller size. We show that statements on the equivalence of the smaller graphs (which are easier to compute) can be used to infer statements on the equivalence of the original, larger graphs.