A generic quantum Wielandt’s inequality

Yifan Jia1,2 and Angela Capel3

1Department of Mathematics, Technische Universität München, Germany
2Munich Center for Quantum Science and Technology (MCQST), Germany
3Fachbereich Mathematik, Universität Tübingen, Germany

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Abstract

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.

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[1] Satvik Singh, Mizanur Rahaman, and Nilanjana Datta, "Zero-error communication, scrambling, and ergodicity", arXiv:2402.18703, (2024).

[2] Jing Bai, Jianquan Wang, and Zhi Yin, "Primitivity for random quantum channels", Quantum Information Processing 23 2, 47 (2024).

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