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

We study the problem of finding elements in the intersection of an arbitrary conic variety in $\mathbb{F}^{n}$ with a given linear subspace (where $\mathbb{F}$ 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.In this work, we propose and analyze an algorithm for solving this problem. Surprisingly, despite the above hardness results we show that our algorithm solves this problem efficiently for “typical” subspaces. Here, the subspace $\mathcal{U} \subseteq \mathbb{F}^{n}$ is chosen generically of a certain dimension, potentially with some generic elements of the variety contained in it. Our main result is a guarantee that our algorithm recovers all the elements of $\mathcal{U}$ that lie in the variety, under some mild non-degeneracy assumptions on the variety. As corollaries, we obtain the following new results:•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.•Uniqueness results and polynomial time algorithmic guarantees 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 $\sum_{i=1}^{R} v_{i} \otimes w_{i}$, where the $v_{i}$ are elements of the given variety $\mathcal{X}$. This implies new uniqueness results and genericity guarantees even in the special case of tensor decompositions.