最佳可分态的一种简单算法
A Simple Algorithm for Best Separable State
AI总结：
针对最佳可分态问题，本文提出一种更简单的SoS松弛取整算法，优化了运行时间，还证明了具有独立研究价值的钉扎引理新变体。
AI中文摘要：
我们研究最佳可分态问题（BSS），该问题旨在求量子测量对非纠缠态的最大接受概率。以经典术语表述，目标是在单位向量x、y上最大化⟨(x⊗y), M(x⊗y)⟩，其中0 ⪯ M ⪯ I，我们将该值记为BSS(M)。我们在“完美完备性” regime下研究BSS：给定满足BSS(M)=1的M，目标是找到最优解x、y——这推广了在保证包含秩1矩阵的ℝ^{n×n}子空间中，寻找与给定子空间最接近的秩1矩阵的问题。该问题已知的最强算法保证有：(1) Barak、Kothari和Steurer提出的算法，可在时间exp(√n (log n)^{O(1)} / ε²)内找到值为1−ε的解；(2) Bhattiprolu、Ghosh、Guruswami、Lee和Tulsiani提出的算法，可在约n^{O(q)}的时间内找到值为q/n的解。我们提出了一种更简单的SoS松弛取整方法，推广了经典的“全局关联取整”技术，并获得了更优的运行时间。对于满足BSS(M)=1的M，我们的算法可在时间n^{O(√n/ε)}内找到值为1−ε的解，在时间n^{O(√q)}内找到值为q/n的解。利用相同技术，我们证明了“钉扎引理”的新变体，这是一种广泛应用于LP/SDP取整、高维概率和统计物理的测度分解定理，我们认为该定理具有独立的研究价值。
英文摘要：
We study the best separable state problem (BSS), which asks for the maximum acceptance probability of a quantum measurement over unentangled states. In classical terms, the goal is to maximize $\langle(x \otimes y), M (x \otimes y)\rangle$ over unit vectors $x,y$ where $0 \preceq M \preceq I$; we call this value $\mathrm{BSS}(M)$. We study $\mathrm{BSS}$ in the "perfect completeness" regime, where given $M$ such that $\mathrm{BSS}(M) = 1$ the goal is to find the best possible solution $x,y$ -- this generalizes the problem of finding a rank-one matrix as close as possible to a given subspace of $\mathbb{R}^{n \times n}$ guaranteed to contain a rank-one matrix. The strongest known algorithmic guarantees for this problem are: (1) an algorithm which finds a solution with value $1-\varepsilon$ in time $\exp(\sqrt{n} (\log n)^{O(1)} / \varepsilon^2)$, due to Barak, Kothari, and Steurer, and (2) an algorithm which finds a solution with value $q/n$ in time roughly $n^{O(q)}$, due to Bhattiprolu, Ghosh, Guruswami, Lee, and Tulsiani. We give a much simpler approach to rounding the SoS relaxation, generalizing the canonical "global correlation rounding" technique, and obtain a better running time. Given $M$ with $\mathrm{BSS}(M) = 1$, our algorithm finds a solution with value $1-ε$ in time $n^{O(\sqrt{n/\varepsilon})}$, and a solution of value $q/n$ in time $n^{O(\sqrt q)}$. Using the same techniques, we prove a new variant of the "pinning lemma", a measure-decomposition theorem widely used in LP/SDP rounding, high-dimensional probability, and statistical physics, which we believe is of independent interest.
