Making Almost Commuting Matrices Commute

Abstract

Suppose two Hermitian matrices A, B almost commute (${\Vert [A,B] \Vert \leq \delta}$). Are they close to a commuting pair of Hermitian matrices, A′, B′, with ${\Vert A-A' \Vert,\Vert B-B'\Vert \leq \epsilon}$ ? A theorem of H. Lin [3] shows that this is uniformly true, in that for every ${\epsilon > 0}$ 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 ${\epsilon}$ . We give uniform bounds relating δ and ${\epsilon}$ . The proof is constructive, giving an explicit algorithm to construct A′ and B′. We provide tighter bounds in the case of block tridiagonal and tridiagonal matrices. 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.