Citation: Kitaev, Alexei. "Almost-idempotent quantum channels and approximate C^*-algebras." arXiv preprint arXiv:2405.02434 (2024).
Web: https://arxiv.org/abs/2405.02434
Tags: Mathematical, MTC-reconstruction, Operator-algebras
In this paper, Kitaev proves that every algebra which approximately satisfies the axioms of a C*-algebra is approximately isomorphic to an actual C*-algebra. He defines a notion of epsilon C*-algebra to be an algebra which satisfies the axioms of a C*-algebra but with associativity and unit axioms only holding with size epsilon corrections. He then defines a notion of a delta-homomorphism to be a map which preserves multiplication up to size delta corrections. The claim is that every epsilon C*-algebra is O(epsilon) isomorphic to a standard C*-algebra.
The utility of this result in entanglement bootstrap and MTC reconstruction is clear. We want to extract the algebra of logical operators from a noisy state. We will only ever be able to extract a noisy version of this algebra. The theorem in question tells us that we can remove the noise. We can find an actual C* algebra which is close to the noisy one we extracted. The proof is nonconstructive, but there are ways of making it constructive that are now known (but unpublished).
A good example of a paper which uses this C*-algebra approach studying topological order is this one, which does a great in-depth study of the toric code:
> Naaijkens, Pieter. "Localized endomorphisms in Kitaev's toric code on the plane." Reviews in Mathematical Physics 23.04 (2011): 347-373.
The proof is very deep. It uses an incredible mix of analysis, cohomology, and operator algebras which I wont begin to describe here. At the end of the day, this paper proves a fundamental theorem which makes this newly-defined notion of C*-algebras a practical one for everyday use.