Making Almost Commuting Matrices Commute

Abstract

Suppose two Hermitian matrices A, B almost commute ( [A, B] ≤ δ). Are they close to a commuting pair of Hermitian matrices, A , B , with A − A , B − B ≤ ? A theorem of H. Lin [3] shows that this is uniformly true, in that for every > 0 there exists a δ > 0, independent of the size N of the matrices, for which almost commuting implies being close to a commuting pair. However, this theorem does not specify how δ depends on . We give uniform bounds relating δ and . We provide tighter bounds in the case of block tridiagonal and tridiagonal matrices and a fully constructive method in that case. Within the context of quantum measurement, this implies an algorithm to construct a basis in which we can make a projective measurement that approximately measures two approximately commuting operators simultaneously. Finally, we comment briefly on the case of approximately measuring three or more approximately commuting operators using POVMs (positive operator-valued measures) instead of projective measurements.

Disordered topological insulators via C * -algebras

“…In this case, there is no topological obstruction to approximating almost commuting matrices by exactly commuting matrices [15,16]; physically, we understand this as if the system on a disk is in a topologically nontrivial phase then there will be gapless boundary modes and hence P XP, P Y P will not almost commute.…”

Essentially Normal Operators

“…Finally we mention that a paper has recently been posted on the arXiv by Hastings [29] which claims a constructive proof that almost commuting Hermitian matrices are close to commuting, with explicit estimates. This is a welcome addition since the soft proof provides no norm estimates at all.…”

Almost commuting unitaries with spectral gap are near commuting unitaries

“…Such a result shouldn't depend on the presence of a gap in the spectrum of U and V . However, at the current time, it is unclear how to use the arguments of Hastings in [8] to prove such a quantitative version.…”

“…The proof was nonconstructive, and the dependence of δ on was not quantified, apart from the nontrivial fact that δ did not depend on the size of the matrices. Recently, Hastings [8] presented a new constructive proof of the result of Lin and was able to calculate quantitative bounds.…”

“…In light of the recent results of [8] it is worthwhile to reconsider the problem of when a pair of almost commuting unitary matrices are near a pair of commuting unitaries. There have been several studies of this problem, partially motivated by Voiculescu's original counterexample, and it was quickly realised that there was, in general, a K-theoretic obstruction [4,5].…”