A Generic Quantum Wielandt’s Inequality

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Yifan Jia^1,2 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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Featured image: Sketch of the proof for Theorem 1, a generic quantum Wielandt’s inequality.

► BibTeX data

@article{Jia2024genericquantum, doi = {10.22331/q-2024-05-02-1331}, url = {https://doi.org/10.22331/q-2024-05-02-1331}, title = {A generic quantum {W}ielandt’s inequality}, author = {Jia, Yifan and Capel, Angela}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {8}, pages = {1331}, month = may, year = {2024} }

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Cited by

[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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