Computing linear sections of varieties: quantum entanglement, tensor decompositions and beyond
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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 ∑_i=1^R v_i ⊗ w_i, where the v_i 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.
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