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Quantum Physics
Title: A generic quantum Wielandt's inequality
(Submitted on 19 Jan 2023 (v1), last revised 25 Apr 2024 (this version, v3))
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.
Submission history
From: Yifan Jia [view email][v1] Thu, 19 Jan 2023 18:57:32 GMT (659kb,D)
[v2] Thu, 27 Jul 2023 18:29:41 GMT (881kb,D)
[v3] Thu, 25 Apr 2024 13:56:41 GMT (810kb,D)
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