Almost commuting unitaries with spectral gap are near commuting unitaries

2008/09/03 by Osborne, Tobias J.
#15A15 #15A27 #47A55 #47B47 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.0809.0602

Abstract

Let M_n be the collection of n x n complex matrices equipped with operator norm. Suppose U, V ∈ M_n are two unitary matrices, each possessing a gap larger than Δin their spectrum, which satisfy ||UV-VU|| ≤ ε. Then it is shown that there are two unitary operators X and Y satisfying XY-YX = 0 and ||U-X|| + ||V-Y|| ≤ E(Δ²/ε) (ε/Δ²)^(1/6), where E(x) is a function growing slower than x^(1/k) for any positive integer k.

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