Constructing Nearby Commuting Matrices for Reducible Representations of $su(2)$ with an Application to Ogata's Theorem

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 $N \to \infty$. In this paper, 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.