# Entanglement and the truncated moment problem

## TL;DR

This paper links quantum entanglement detection to the truncated moment problem, providing a hierarchy of semi-definite programs for certifying separability or entanglement with explicit decompositions.

## Contribution

It introduces a novel approach that maps entanglement detection to the truncated moment problem, enabling a systematic and computationally feasible certification process.

## Key findings

- Algorithm certifies entanglement with a certificate if entangled.
- Finite steps typically produce a certificate of separability.
- Explicit separable decompositions can be extracted when states are separable.

## Abstract

We map the quantum entanglement problem onto the mathematically well-studied truncated moment problem. This yields a necessary and sufficient condition for separability that can be checked by a hierarchy of semi-definite programs. The algorithm always gives a certificate of entanglement if the state is entangled. If the state is separable, typically a certificate of separability is obtained in a finite number of steps and an explicit decomposition into separable pure states can be extracted.