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. Surprisingly, despite these hardness results we develop an algorithm that solves this problem efficiently for "typical" subspaces. Here, the subspace $U \subseteq \mathbb{F}^n$ 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 $U$ that lie in the variety, under some mild non-degeneracy assumptions on the variety. As corollaries, we obtain the following new results: $\bullet$ 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. $\bullet$ 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 variety $X$. This implies new uniqueness results and genericity guarantees even in the special case of tensor decompositions.

翻译：我们研究在任意圆锥簇（定义于$\mathbb{F}^n$中，其中$\mathbb{F}$可为实数或复数域）与给定线性子空间的交集中寻找元素的问题。该问题通过选择不同代数簇，涵盖了一系列丰富的算法问题。当代数簇特化为秩1矩阵时，已与量子信息论、张量分解等不同领域的核心问题产生深刻关联。尽管该问题在最坏情况下已知为NP难（即使对于秩1矩阵簇也是如此），令人惊讶的是，我们仍能针对"典型"子空间开发出高效求解算法。此处，子空间$U \subseteq \mathbb{F}^n$按特定维度取一般性选择，并可能包含代数簇中的若干一般性元素。我们的主要结果是在代数簇满足某些弱非退化假设的条件下，保证所提算法能恢复$U$中位于该代数簇内的全部元素。作为推论，我们获得以下新成果：$\bullet$ 针对量子纠缠中的多个纠缠子空间问题（包括判定子空间的r-纠缠、完全纠缠与真纠缠）的多项式时间算法。尽管这些问题在最坏情况下均为NP难，但对于维数不超过理论最大值的常数倍的一般性子空间，我们的求解算法可在多项式时间内完成。$\bullet$ 对超越张量分解的广泛低秩分解问题的一般性实例，给出唯一性结果及多项式时间算法保证。在此框架下，我们可恢复形如$\sum_{i=1}^R v_i \otimes w_i$的分解（其中$v_i$属于代数簇$X$）。即使限定于张量分解的特例，该结果也蕴含新颖的唯一性结论与一般性保证。