Suppose two Hermitian matrices $A,B$ almost commute ($\Vert [A,B] \Vert \leq ��$). Are they close to a commuting pair of Hermitian matrices, $A',B'$, with $\Vert A-A' \Vert,\Vert B-B'\Vert \leq ��$? A theorem of H. Lin shows that this is uniformly true, in that for every $��>0$ there exists a $��>0$, 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 $��$ depends on $��$. We give uniform bounds relating $��$ 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 {\it 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.

22 pages; tighter bounds; Note: fixed mistake in proof pointed out by Filonov and Kachkovskiy