Almost commuting self-adjoint operators and measurements

TL;DR

This paper investigates when almost commuting self-adjoint operators can be approximated by truly commuting ones, providing conditions based on spectral properties and exploring the problem's limitations in higher dimensions.

Contribution

It offers an affirmative condition for approximating almost commuting operators by commuting ones when spectral spectra are close, and clarifies the problem's limitations for higher operator tuples.

Findings

Approximation is possible when synthetic-spectrum and essential synthetic-spectrum are close.

Counterexamples show the approximation fails for n ≥ 3 even with zero Fredholm index.

For n=2, approximation depends on the vanishing of a Fredholm index outside the essential synthetic-spectrum.

Abstract

We study the problem when an almost commuting $n$-tuple self-adjoint operators in an infinite dimensional separable Hilbert space $H$ is close to an $n$-tuple of commuting self-adjoint operators on $H.$ We give an affirmative answer to the problem when the synthetic-spectrum and the essential synthetic-spectrum are close. Examples are also exhibited that, in general, the answer to the problem when $n\ge 3$ is negative even the associated Fredholm index vanishes. In the case that $n=2,$ we show that a pair of almost commuting self-adjoint operators in an infinite dimensional separable Hilbert space is close to a commuting pair of self-adjoint operators if and only if a corresponding Fredholm index vanishes outside of an essential synthetic-spectrum. This is an attempt to solve a problem proposed by David Mumford related to quantum theory and measurements.