## Computing linear sections of varieties: quantum entanglement, tensor decompositions and beyond

**Abstract:**

We study the problem of finding elements in the intersection of an arbitrary conic variety in F^n with a given linear subspace (where F can be the real or complex field). This problem captures a rich family of algorithmic problems under different choices of the variety. The special case of the variety consisting of rank-1 matrices already has strong connections to central problems in different areas like quantum information theory and tensor decompositions. This problem is known to be NP-hard in the worst-case, even for the variety of rank-1 matrices.

Surprisingly, despite these hardness results we give efficient algorithms that solve this problem for “typical” subspaces. Here, the subspace U⊆F^n is chosen generically of a certain dimension, potentially with some generic elements of the variety contained in it. Our main algorithmic result is a polynomial time algorithm that recovers all the elements of U that lie in the variety, under some mild non-degeneracy assumptions on the variety. As corollaries, we obtain the following results:

• Uniqueness results and polynomial time algorithms for generic instances of a broad class of low-rank decomposition problems that go beyond tensor decompositions. Here, we recover a decomposition of the form ∑vi⊗wi, where the vi are elements of the given variety X. This implies new algorithmic results even in the special case of tensor decompositions.

• Polynomial time algorithms for several entangled subspaces problems in quantum entanglement, including determining r-entanglement, complete entanglement, and genuine entanglement of a subspace. While all of these problems are NP-hard in the worst case, our algorithm solves them in polynomial time for generic subspaces of dimension up to a constant multiple of the maximum possible.

**Authors:**

- Nathaniel Johnston
- Benjamin Lovitz
- Aravindan Vijayaraghavan

**Download:**

- Local preprint
- Preprint from arXiv:2212.03851 [quant-ph]

**Cite as:**

- N. Johnston, B. Lovitz, and A. Vijayaraghavan.
*Computing linear sections of varieties: quantum entanglement, tensor decompositions and beyond*. E-print: arXiv:2212.03851 [quant-ph], 2022.

**Related Papers:**

- A Complete Hierarchy of Linear Systems for Certifying Quantum Entanglement of Subspaces – a paper that uses these techniques to focus specifically on entangled subspaces in quantum information theory