Free versus Bound Entanglement: Machine learning tackling a NP-hard problem

TL;DR

This paper uses machine learning to analyze a large family of bipartite qutrit states, revealing that bound entanglement is more common than previously thought and uncovering a low-dimensional structure in bound entangled states.

Contribution

It introduces a new family of symmetric states and applies machine learning to characterize bound entanglement, revealing a low-dimensional structure and its prevalence.

Findings

82% of states are free entangled

10% of states are bound entangled

Strong 2D linear structure in bound entangled states

Abstract

Entanglement detection in high dimensional systems is a NP-hard problem since it is lacking an efficient way. Given a bipartite quantum state of interest free entanglement can be detected efficiently by the PPT-criterion (Peres-Horodecki criterion), in contrast to detecting bound entanglement, i.e. a curious form of entanglement that can also not be distilled into maximally (free) entangled states. Only a few bound entangled states have been found, typically by constructing dedicated entanglement witnesses, so naturally the question arises how large is the volume of those states. We define a large family of magically symmetric states of bipartite qutrits for which we find $82\%$ to be free entangled, $2\%$ to be certainly separable and as much as $10\%$ to be bound entangled, which shows that this kind of entanglement is not rare. Via various machine learning algorithms we can confirm that the remaining $6\%$ of states are more likely to belonging to the set of separable states than bound entangled states. Most important we find via dimension reduction algorithms that there is a strong $2$-dimensional (linear) sub-structure in the set of bound entangled states. This revealed structure opens a novel path to find and characterize bound entanglement towards solving the long-standing problem of what the existence of bound entanglement is implying.