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

**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.

**Authors:**

- Nathaniel Johnston
- Benjamin Lovitz
- Aravindan Vijayaraghavan

**Download:**

- Official publication in Physical Review A
- Local preprint
- Preprint from arXiv:2210.16389 [quant-ph]

**Cite as:**

- N. Johnston, B. Lovitz, and A. Vijayaraghavan. Complete hierarchy of linear systems for certifying quantum entanglement of subspaces.
*Physical Review A*, 106:062443, 2022.

**Supplementary Material:**

- EntSubspaceMATLAB.zip – a collection of MATLAB scripts for numerically implementing the methods of this paper

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- Computing linear sections of varieties: quantum entanglement, tensor decompositions and beyond – a paper that uses similar techniques in a much more general setting to solve tensor decomposition problems