Entanglement of $\pi$-LME states and the SAT problem

In this paper we investigate the entanglement properties of the class of $\pi$-locally maximally entanglable ($\pi$-LME) states, which are also known as the "real equally weighted states" or the "hypergraph states". The $\pi$-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 $\pi$-LME state is entangled would imply an efficient solution to the boolean satisfiability (SAT) 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.