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 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 develop an algorithm that solves this problem efficiently 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 result is a guarantee that our algorithm 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 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 w i, where the v i are elements of the variety X. This implies new uniqueness results and genericity guarantees even in the special case of tensor decompositions.