The multiplicative domain in quantum error correction

TL;DR

This paper introduces a new class of quantum error correcting codes derived from the multiplicative domain of completely positive maps, connecting them to unitarily correctable codes and providing a representation theoretic framework.

Contribution

It characterizes quantum error correcting codes using the multiplicative domain, linking unital channels to measurement-free recovery and extending to non-unital channels with a new algebraic approach.

Findings

Identifies the multiplicative domain as a source of quantum error correcting codes.

Shows unital channels correspond to codes without measurement in recovery.

Provides a representation theoretic characterization of subsystem codes.

Abstract

We show that the multiplicative domain of a completely positive map yields a new class of quantum error correcting codes. In the case of a unital quantum channel, these are precisely the codes that do not require a measurement as part of the recovery process, the so-called unitarily correctable codes. Whereas in the arbitrary, not necessarily unital case they form a proper subset of unitarily correctable codes that can be computed from properties of the channel. As part of the analysis we derive a representation theoretic characterization of subsystem codes. We also present a number of illustrative examples.