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

TL;DR

This paper establishes quantitative bounds on how almost-commuting operator pairs in matrix algebras are close to truly commuting pairs, using probabilistic and deterministic methods, with applications to quantum information theory.

Contribution

It introduces new probabilistic and deterministic formulations of almost-commutation, leading to quantitative Tsirelson's theorems and generalizations of Stampfli's theorem in finite dimensions.

Findings

Operators close to commutants within O(d^2 ε) in operator norm.

Probabilistic generalizations of Stampfli's theorem for Haar-random unitaries.

Approximate quantum models are well-approximated by tensor-product models with error O(d^2 ε).

Abstract

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 $O(d^2\epsilon)$-close in operator norm to an operator in the commutant $\mathcal{A}'$ under two complementary formulations of "$\epsilon$-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 $\epsilon$-almost quantum commuting observable model is well approximated by a quantum tensor-product model with error $O(d^2\epsilon)$.