Making almost commuting matrices commute

Journal Article · Tue Jan 01 04:00:00 EST 2008 · Communications in Mathematical Physics

DOI:https://doi.org/10.1007/s00220-009-0877-2· OSTI ID:956595

Hastings, Matthew B ^[1]

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1. Los Alamos National Laboratory

Suppose two Hermitian matrices A, B almost commute ({parallel}[A,B]{parallel} {<=} {delta}). Are they close to a commuting pair of Hermitian matrices, A', B', with {parallel}A-A'{parallel},{parallel}B-B'{parallel} {<=} {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 specifiy 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 tridiagnonal 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.

• Research Organization: Los Alamos National Laboratory (LANL)
• Sponsoring Organization: DOE
• DOE Contract Number: AC52-06NA25396
• OSTI ID: 956595
• Report Number(s): LA-UR-08-05139; LA-UR-08-5139
• Journal Information: Communications in Mathematical Physics, Journal Name: Communications in Mathematical Physics Journal Issue: 2 Vol. 291; ISSN 0010-3616; ISSN CMPHAY
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ALGORITHMS
HERMITIAN MATRIX
MATRICES
MATRIX ELEMENTS
MEASURING METHODS
SIZE