Making Almost Commuting Matrices Commute TeX Export

@article{citeulike:3138287, 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\cite{hl} 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\$. 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 {\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.}, archivePrefix = {arXiv}, author = {Hastings, M. B.}, citeulike-article-id = {3138287}, citeulike-linkout-0 = {http://arxiv.org/abs/0808.2474}, citeulike-linkout-1 = {http://arxiv.org/pdf/0808.2474}, day = {18}, eprint = {0808.2474}, keywords = {algebra, approximate, commutative, matrix}, month = {Aug}, posted-at = {2008-12-06 08:26:12}, priority = {2}, title = {Making Almost Commuting Matrices Commute}, url = {http://arxiv.org/abs/0808.2474}, year = {2008} }

TY - JOUR ID - citeulike:3138287 L3 - citeulike-article-id:3138287 N2 - 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\cite{hl} 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 $\epsilon$. 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 {\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. TI - Making Almost Commuting Matrices Commute KW - algebra KW - approximate KW - commutative KW - matrix AU - Hastings, MB PY - 2008/Aug/18/ UR - http://arxiv.org/abs/0808.2474 ER -