科学研究
报告题目:

A generic quantum Wielandt's inequality for matrix and Lie algebras

报告人:

贾依凡 博士(Technical University of Munich)

报告时间:

报告地点:

金沙9001cc诚为本东北楼二楼报告厅(209)

报告摘要:

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.