Making Almost Commuting Matrices Commute

Abstract

Suppose two Hermitian matrices $A,B$ almost commute ($\Vert [A,B]\Vert \le \delta$). Are they close to a commuting pair of Hermitian matrices, $A^{\prime },B^{\prime }$, with $\Vert A-A^{\prime }\Vert ,\Vert B-B^{\prime }\Vert \le \epsilon$? 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.

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

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Cited by in corpus (35)

• Disordered Topological Insulators via $C^{\ast }$-Algebras
• Long-Time Behavior of Macroscopic Quantum Systems: Commentary Accompanying the English Translation of John von Neumann's 1929 Article on the Quantum Ergodic Theorem
• K-Theory and Pseudospectra for Topological Insulators
• Topological Insulators and C^*-Algebras: Theory and Numerical Practice
• Almost Commuting Matrices, Localized Wannier Functions, and the Quantum Hall Effect
• Classicality, Markovianity and local detailed balance from pure state dynamics
• Almost commuting matrices with respect to normalized Hilbert-Schmidt norm
• Quasi-conserved charges in the perturbed spin-1/2 XXX model
• Detecting Topological Order with Ribbon Operators
• First Principles Numerical Demonstration of Emergent Decoherent Histories
• Local formula for the $Z_2$ invariant of topological insulators
• Quasi-morphisms and quasi-states in symplectic topology
• Abelian groups are polynomially stable
• A Hilbert-Schmidt analog of Huaxin Lin's Theorem
• Asymptotically Decreasing Lieb-Robinson Velocity for a Class of Dissipative Quantum Dynamics
• Distance to normal elements in $C^{\ast }$-algebras of real rank zero
• Misanthropic Entropy and Renormalization as a Communication Channel
• Blocks in cycles and k-commuting permutations
• Approximate symmetries of Hamiltonians
• Almost commuting self-adjoint matrices --- the real and self-dual cases
• Almost-commuting matrices are almost jointly diagonalizable
• Algebraic localization implies exponential localization in non-periodic insulators
• Quasi-conserved quantities in the perturbed XXX spin chain
• Inhomogeneous SU(2) symmetries in homogeneous integrable U(1) circuits and transport
• Dynamical typicality for initial states with a preset measurement statistics of several commuting observables
• On the relation between an operator and its self-commutator
• Certification of two-qubit quantum systems with temporal inequality
• Complex-Time Singularity and Locality Estimates for Quantum Lattice Systems
• Stationarity of quantum statistical ensembles at first-order phase transition points
• Almost commuting matrices and stability for product groups
• Statistical ensembles for phase coexistence states specified by noncommutative additive observables
• Asymptotic metrics on the space of matrices under the commutation relation
• On-State Commutativity of Measurements and Joint Distributions of Their Outcomes
• Poisson brackets and symplectic invariants
• Almost commuting unitaries with spectral gap are near commuting unitaries