Quantum error correction on infinite-dimensional Hilbert spaces

TL;DR

This paper extends quantum error correction theory to infinite-dimensional Hilbert spaces, introducing new classes of codes based on von Neumann algebras, broadening the scope beyond finite-dimensional systems.

Contribution

It generalizes quantum error correction to infinite-dimensional spaces using algebraic approaches, enabling correction of codes defined by infinite-dimensional von Neumann algebras.

Findings

Introduces a framework for quantum error correction on infinite-dimensional Hilbert spaces.

Identifies new classes of quantum error correcting codes without finite-dimensional counterparts.

Demonstrates examples of codes defined by infinite-dimensional algebras.

Abstract

We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. The generalization yields new classes of quantum error correcting codes that have no finite-dimensional counterparts. The error correction theory we develop begins with a shift of focus from states to algebras of observables. Standard subspace codes and subsystem codes are seen as the special case of algebras of observables given by finite-dimensional von Neumann factors of type I. Our generalization allows for the correction of codes characterized by any von Neumann algebra and we give examples, in particular, of codes defined by infinite-dimensional algebras.