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

TL;DR

This paper explores the entanglement properties of $ ext{pi}$-LME states, revealing that efficiently detecting their entanglement is as hard as solving NP-complete SAT problems, highlighting the complexity of entanglement detection.

Contribution

It proves that deciding entanglement in $ ext{pi}$-LME states is NP-complete, linking quantum entanglement detection to classical computational complexity.

Findings

Entanglement detection in $ ext{pi}$-LME states is NP-complete.

The problem remains hard even for highly symmetric states.

Efficient entanglement detection would imply an efficient SAT solution.

Abstract

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.