Quantum Wielandt's inequality gives an optimal upper bound on the minimal length $k$ such that length-$k$ products of elements in a generating system span $M_n(\mathbb{C})$. It is conjectured that $k$ should be of order $\mathcal{O}(n^2)$ in general. In this paper, we give an overview of how the question has been studied in the literature so far and its relation to a classical question in linear algebra, namely the length of the algebra $M_n(\mathbb{C})$. We provide a generic version of quantum Wielandt's inequality, which gives the optimal length with probability one. More specifically, we prove based on [KS16] that $k$ generically is of order $\Theta(\log n)$, as opposed to the general case, in which the best bound to date is $\mathcal O(n^2 \log n)$. Our result implies a new bound on the primitivity index of a random quantum channel. Furthermore, we shed new light on a long-standing open problem for Projected Entangled Pair State, by concluding that almost any translation-invariant PEPS (in particular, Matrix Product State) with periodic boundary conditions on a grid with side length of order $\Omega( \log n )$ is the unique ground state of a local Hamiltonian. We observe similar characteristics for matrix Lie algebras and provide numerical results for random Lie-generating systems.

中文翻译：

---

通用量子维兰特不等式

量子维兰特不等式给出了最小长度 $k$ 的最佳上限，使得生成系统中元素的长度 $k$ 乘积跨越 $M_n(\mathbb{C})$。据推测，$k$ 一般应为 $\mathcal{O}(n^2)$ 阶。在本文中，我们概述了迄今为止文献中如何研究该问题及其与线性代数中的经典问题（即代数 $M_n(\mathbb{C})$ 的长度）的关系。我们提供了量子维兰特不等式的通用版本，它给出了概率为 1 的最佳长度。更具体地说，我们根据 [KS16] 证明 $k$ 通常具有 $\Theta(\log n)$ 的阶数，这与一般情况相反，在一般情况下，迄今为止的最佳边界是 $\mathcal O(n^ 2 \log n)$。我们的结果暗示了随机量子通道的原始指数的新界限。此外，我们得出的结论是，几乎所有平移不变的 PEPS（特别是矩阵乘积状态）在边长为 $\ 的网格上都具有周期性边界条件，从而对投影纠缠对状态的长期悬而未决的问题有了新的认识Omega( \log n )$ 是局部哈密顿量的唯一基态。我们观察到矩阵李代数的类似特征，并提供随机李生成系统的数值结果。