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 talk, we first 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 matrix algebra $M_n(\mathbb{C})$. We provide a generic version of Quantum Wielandt's inequality, which gives the optimal length with probability one. Specifically, we prove, based on [Klep and Spenko, 16] that $k$ generically is of order $\Theta(\log n)$, as opposed to the general case, where 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.

Additionally, we observe similar characteristics for matrix Lie algebras, which we define as the Lie-length of a matrix Lie algebra. Numerical results for random Lie-generating systems indicate that Lie-lengths generically scale as $\Theta(\log n)$ for some typical $\Theta(n^2)$-dimensional Lie algebras e.g. $\mathfrak{su}(n)$, with a small number of Lie-generators. We prove this upper bound for almost any Lie-generating system $S\subseteq sl_n(\mathbb{C})$ with $\Omega(\log n)$ elements and indicate how this generic result for Lie algebras would apply to mathematical models of some physical problems, especially also to gate implementation of quantum computers.

报告人简介：Dr. Yifan Jia is from Technical University of Munich, working under the supervision of Prof. Michael Wolf. She is broadly interested in quantum information theory, with a particular focus on circuit complexity and complexity notions in quantum many-body systems. She has spoken at the top quantum information conference QIP2023 and has publication in Comm. Math. Phys. In August 2024, she will begin a new position as a postdoctoral researcher at the University of Copenhagen.