For every field F, every integer n≥ 1, and every family S⊆ M_n(F), if there exists an integer ell≥ 1 such that operatornamespan_FA_1A_2… A_ell:A_i∈ S=M_n(F), then for every integer k≥ n^2+2n-3 one has operatornamespan_FA_1A_2… A_k:A_i∈ S=M_n(F).

Pérez-García–Verstraete–Wolf–Cirac conjecture

The conjecture asks whether full matrix-algebra growth, once reached, persists after a bound proportional to the square of the matrix dimension. Earlier work reduced the known bound substantially, but the exact claimed order was not reached.

Known results

For a matrix space $L$ of $D\times D$ matrices, earlier work had an $O(D^4)$ bound; a later theorem gives $\mathcal{I}\le 2D^2(6+\log_2D)$ , confirming the conjectured exponent $2$ but retaining a logarithmic factor.

September 3, 2026 journal paper

On September 3, 2026, Yaroslav Shitov’s paper Growth in Matrix Algebras and a Conjecture of Pérez-García, Verstraete, Wolf and Cirac appeared online in Annales Henri Poincaré. It reports new analysis of the conjecture, but the retrieved record gives no abstract or theorem statement, so its precise implication is unverified.

Current status (as of September 2026): the conjecture has known $O(D^2\log D)$ -type progress, while the new paper’s exact result and whether it establishes the conjectured $O(D^2)$ bound remain undetermined.

Pérez-García–Verstraete–Wolf–Cirac conjecture

Primary source

Additional references

Progress summary

Solutions 0