On the local equivalence of complete bipartite and repeater graph states

TL;DR

This paper investigates the local equivalence of certain quantum graph states, demonstrating that complete bipartite, imperfect repeater, and small 'crazy' graph states satisfy LU $\\Leftrightarrow$ LC, with implications for quantum repeater protocols.

Contribution

It establishes LU $\\Leftrightarrow$ LC for specific classes of graph states relevant to quantum repeaters and discusses their relation to the LU-LC Conjecture, providing new proofs and insights.

Findings

Complete bipartite graph states satisfy LU $\\Leftrightarrow$ LC.

Imperfect repeater and small 'crazy' graph states satisfy LU $\\Leftrightarrow$ LC.

Discussion of biclique states in the context of LU-LC Conjecture counterexamples.

Abstract

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.