Entanglement ofπ–locally-maximally-entangleable states and the satisfiability problem

Abstract

In this paper we investigate the entanglement properties of the class of $\ensuremath{\pi}$--locally-maximally-entangleable $(\ensuremath{\pi}$-LME) states, which are also known as the real equally weighted states or the hypergraph states. The $\ensuremath{\pi}$-LME states comprise well-studied classes of quantum states (e.g., graph states) and exhibit a large degree of symmetry. Motivated by the structure of LME states, we show that the capacity to (efficiently) determine if a $\ensuremath{\pi}$-LME state is entangled would imply an efficient solution to the Boolean satisfiability problem. More concretely, we show that this particular problem of entanglement detection, phrased as a decision problem, is $\mathsf{NP}$-complete. The restricted setting we consider yields a technically uninvolved proof, and illustrates that entanglement detection, even when quantum states under consideration are highly restricted, still remains difficult.