On the local equivalence of complete bipartite and repeater graph states

Classifying locally equivalent graph states, and stabilizer states more broadly, is a significant problem in the theories of quantum information and multipartite entanglement. A special focus is given to those graph states for which equivalence through local unitaries implies equivalence through local Clifford unitaries (LU $\Leftrightarrow$ LC). Identification of locally equivalent states in this class is facilitated by a convenient transformation rule on the underlying graphs and an efficient algorithm. Here we investigate the question of local equivalence of the graph states behind the all-photonic quantum repeater. We show that complete bipartite graph (biclique) states, imperfect repeater graph states and small "crazy graph" states satisfy LU $\Leftrightarrow$ LC. We continue by discussing biclique states more generally and placing them in the context of counterexamples to the LU-LC Conjecture. To this end, we offer some alternative proofs and clarifications on existing results.