Making Almost Commuting Matrices Commute

TL;DR

This paper establishes uniform bounds relating how closely almost commuting Hermitian matrices can be approximated by truly commuting matrices, with applications to quantum measurement and constructive algorithms.

Contribution

It provides explicit bounds on the relationship between almost commutation and proximity to commuting matrices, including tighter bounds for special matrix classes and a constructive method.

Findings

Uniform bounds relating $ orm{[A,B]}$ and distance to commuting matrices

Tighter bounds for block tridiagonal and tridiagonal matrices

A constructive algorithm for approximately commuting matrices in quantum measurement

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 shows that this is uniformly true, in that for every $\epsilon>0$ there exists a $\delta>0$, 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 $\delta$ depends on $\epsilon$. We give uniform bounds relating $\delta$ and $\epsilon$. 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.