Almost commuting unitaries with spectral gap are near commuting unitaries

TL;DR

This paper proves that almost commuting unitaries with spectral gaps can be approximated by exactly commuting unitaries, with the approximation quality depending on the spectral gap and the commutator norm.

Contribution

It establishes a quantitative approximation result for almost commuting unitaries with spectral gaps by exactly commuting unitaries, extending previous results.

Findings

Approximate unitaries are close to commuting unitaries when spectral gaps are present.

The approximation error depends on the spectral gap and the commutator norm.

The function controlling the approximation error grows slower than any fractional power of the inverse of the commutator norm.

Abstract

Let M_n be the collection of n x n complex matrices equipped with operator norm. Suppose U, V \in M_n are two unitary matrices, each possessing a gap larger than \Delta in their spectrum, which satisfy ||UV-VU|| \le \epsilon. Then it is shown that there are two unitary operators X and Y satisfying XY-YX = 0 and ||U-X|| + ||V-Y|| \le E(\Delta^2/\epsilon) (\epsilon/\Delta^2)^(1/6), where E(x) is a function growing slower than x^(1/k) for any positive integer k.