Let S=v_i_i=1^g⊆matrices be a collection of matrices, and define its Wielength by operatornameWieell(S):=infk∈N:operatornamespanS^k=matrices, where S^k=v_sigma:sigma∈1,…,g^k, v_sigma=v_i_k… v_i_1 for sigma=(i_1,…,i_k), and matrices denotes the relevant matrix space. Quantum Wielandt conjecture. For all such S, operatornameWieell(S)≤ O(D^2). This conjecture concerns the length needed for products of a collection of matrices to linearly span the full matrix space and is motivated by its relationship with parent Hamiltonians in matrix product states. The source relates it to Conjecture 2 of Pérez-García et al.; no resolution is given here.

The quantum Wielandt conjecture for matrix sets

From papers

Let $S=\{v_i\}_{i=1}^g\subseteq\matrices$ be a collection of matrices, and define its Wielength by

\operatorname{Wie}\ell(S):=\inf\{k\in\mathbb{N}:\operatorname{span}S^k=\matrices\},

where $S^k=\{v_\sigma:\sigma\in\{1,\dots,g\}^k\}$ , $v_\sigma=v_{i_k}\cdots v_{i_1}$ for $\sigma=(i_1,\dots,i_k)$ , and $\matrices$ denotes the relevant matrix space. Quantum Wielandt conjecture. For all such $S$ ,

\operatorname{Wie}\ell(S)\leq O(D^2).

This conjecture concerns the length needed for products of a collection of matrices to linearly span the full matrix space and is motivated by its relationship with parent Hamiltonians in matrix product states. The source relates it to Conjecture 2 of Pérez-García et al.; no resolution is given here.

Progress summary

Solved

A 2023 preprint claims the conjecture is settled, but later research still treats the general case as open and no independent verification was found.

The conjecture, associated with Conjecture 2 of Pérez-García, Verstraete, Wolf, and Cirac, asks for a universal quadratic bound $\operatorname{Wie}\ell(S)=O(D^2)$ on the length needed for matrix products to span the full matrix space.

Known results

Primitive quantum channels: bound $q\leq (D^2-d+1)D^2$ (2009).
General linear-space formulation: $O(D^2\log D)$ , improving $O(D^4)$ (2018).
Generic matrix sets: injectivity at logarithmic length, while the worst-case bound remains $O(D^2\log D)$ (2024).

August 2023 claim; 2024–2025 status

A viXra preprint claims a strong resolution, asserting $w(n)\leq n^2+2n-4$ for $n>2$ . However, the 2024 paper explicitly says the general conjecture remains open, and a 2025 note discusses only a possible approach; no independent verification or accepted proof was found.

Current status (as of August 2026): The best established general bound is $O(D^2\log D)$ , the generic case is much better, and the universal $O(D^2)$ conjecture remains open despite an unverified 2023 claim.

Sources

Sources & referencesView supporting material

Primary source

Owen Ekblad, “A note on the quantum Wielandt inequality”, arXiv:2504.21638 (2025).

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