A Complete Hierarchy of Linear Systems for Certifying Quantum Entanglement of Subspaces

TL;DR

This paper presents a new hierarchy of linear systems that efficiently certifies entanglement in quantum subspaces, outperforming existing methods and applicable to complex, large-scale quantum systems.

Contribution

It introduces a complete, scalable hierarchy of linear systems for certifying entanglement, generalizable to higher Schmidt rank and multipartite systems, using elementary linear algebra techniques.

Findings

Outperforms known methods at the first hierarchy level

Complete in certifying all entangled subspaces

Efficient implementation for large quantum systems

Abstract

We introduce a hierarchy of linear systems for showing that a given subspace of pure quantum states is entangled (i.e., contains no product states). This hierarchy outperforms known methods already at the first level, and it is complete in the sense that every entangled subspace is shown to be so at some finite level of the hierarchy. It generalizes straightforwardly to the case of higher Schmidt rank, as well as the multipartite cases of completely and genuinely entangled subspaces. These hierarchies work extremely well in practice even in very large quantum systems, as they can be implemented via elementary linear algebra techniques rather than the semidefinite programming techniques that are required by previously-known hierarchies.