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.

We prove limitations on LOCC and separable measurements in bipartite state discrimination problems using techniques from convex optimization. Specific results that we prove include: an exact formula for the optimal probability of correctly discriminating any set of either three or four Bell states via LOCC or separable measurements when the parties are given an ancillary partially entangled pair of qubits; an easily checkable characterization of when an unextendable product set is perfectly discriminated by separable measurements, along with the first known example of an unextendable product set that cannot be perfectly discriminated by separable measurements; and an optimal bound on the success probability for any LOCC or separable measurement for the recently proposed state discrimination problem of Yu, Duan, and Ying.