A unification of the coding theory and OAQEC perspective on hybrid codes

TL;DR

This paper unifies the coding theory and operator algebra quantum error correction perspectives on hybrid codes, demonstrating their equivalence, generalizing bounds, and constructing a non-trivial example.

Contribution

It shows the equivalence of two hybrid code characterizations, generalizes the quantum Hamming bound, and constructs a new non-trivial hybrid code example.

Findings

Coding theory and OAQEC perspectives are equivalent.

A generalized quantum Hamming bound for hybrid codes is introduced.

A non-trivial degenerate 4-qubit hybrid code is constructed.

Abstract

There is an advantage in simultaneously transmitting both classical and quantum information over a quantum channel compared to sending independent transmissions. The successful implementation of simultaneous transmissions of quantum and classical information will require the development of hybrid quantum-classical error-correcting codes, known as hybrid codes. The characterization of hybrid codes has been performed from a coding theory perspective and an operator algebra quantum error correction (OAQEC) perspective. First, we demonstrate that these two perspectives are equivalent and that the coding theory characterization is a specific case of the OAQEC model. Second, we include a generalization of the quantum Hamming bound for hybrid error-correcting codes. We discover a necessary condition for developing non-trivial hybrid codes -- they must be degenerate. Finally, we construct an example of a non-trivial degenerate 4-qubit hybrid code.