Quantitative Tsirelson's Theorems via Approximate Schur's Lemma and Probabilistic Stampfli's Theorems

Whether an almost-commuting pair of operators must be close to a commuting pair is a central question in operator and matrix theory. We investigate this problem for pairs of $C^*$-subalgebras $\mathcal{A}$ and $\mathcal{B}$ of $M_d(\mathbb{C})$, showing that each operator in $\mathcal{B}$ is $O(d^2\varepsilon)$-close in operator norm to an operator in the commutant $\mathcal{A}'$ under two complementary formulations of ``$\varepsilon$-almost commutation.’' One formulation is probabilistic, requiring that the operators of $\mathcal{B}$ have small commutators for most Haar-random unitaries acting on $\mathcal{A}$. This first formulation leads to two novel probabilistic generalizations of Stampfli’s theorem, which relates an operator’s distance from the scalars to the norm of its inner derivation. The second formulation is deterministic, requiring small commutators between the generators of $\mathcal{A}$ and $\mathcal{B}$; we analyze this using an approximate Schur’s lemma formulated in terms of Weyl-Heisenberg (clock-and-shift) matrices. As an application of our results to quantum information theory, we obtain a quantitative Tsirelson’s theorem: in dimension $d$, every $\varepsilon$-almost quantum commuting observable model is well approximated by a quantum tensor-product model with error $O(d^2\varepsilon)$.