Constructing Nearby Commuting Matrices for Ogata's Theorem on Macroscopic Observables

TL;DR

This paper constructs explicit nearby commuting matrices for macroscopic observables in quantum systems, proving Ogata's theorem for site dimension 2 and exploring real observables' asymptotic commutativity.

Contribution

It provides a constructive method to approximate non-commuting observables with commuting ones in large quantum systems, specifically for $su(2)$ representations.

Findings

Explicit estimates for proximity of commuting matrices in Ogata's theorem

Construction of real commuting observables close to real macroscopic observables

Proof of Ogata's theorem for site dimension d=2

Abstract

Resolving a conjecture of von Neumann, Ogata's theorem in arXiv:1111.5933 showed the highly nontrivial result that arbitrarily many matrices corresponding to macroscopic observables with $N$ sites and a fixed site dimension $d$ are asymptotically nearby commuting observables as $N \to \infty$. We develop a method to construct nearby commuting matrices for normalized highly reducible representations of $su(2)$ whose multiplicities of irreducible subrepresentations exhibit a certain monotonically decreasing behavior. We then provide a constructive proof of Ogata's theorem for site dimension $d=2$ with explicit estimates for how close the nearby observables are. Moreover, motivated by the application to time-reversal symmetry explored in arXiv:1012.3494, our construction has the property that real macroscopic observables are asymptotically nearby real commuting observables. This thesis also contains an introduction to the prerequisite matrix analysis for understanding the proof of the results gotten, a primer on the functional analysis of $C^\ast$-algebras frequently used in the literature of almost commuting operators, a detailed discussion of the history and main results in the problem of almost commuting matrices, a review of the basics of measurement of observables in finite dimensional quantum mechanical systems, and a discussion of the uncertainty principle for non-commuting bounded operators with some results obtained for observables with nearby commuting approximants.