Generalized Perfect Codes for Symmetric Classical-Quantum Channels

TL;DR

This paper introduces a new family of codes for symmetric classical-quantum channels, extending classical perfect codes to quantum settings, and demonstrates their optimality and minimal error probability for specific channels.

Contribution

It generalizes classical perfect codes to quantum channels, establishing optimality conditions and identifying Bell state-based codes as quasi-perfect for certain channels.

Findings

Codes based on generalized Bell states are quasi-perfect for some N-qubit channels.

These codes achieve the smallest error probability among all codes of same size and blocklength.

Optimality depends on the channel and an auxiliary state in the output space.

Abstract

We define a new family of codes for symmetric classical-quantum channels and establish their optimality. To this end, we extend the classical notion of generalized perfect and quasi-perfect codes to channels defined over some finite dimensional complex Hilbert output space. The resulting optimality conditions depend on the channel considered and on an auxiliary state defined on the output space of the channel. For certain $N$-qubit classical-quantum channels, we show that codes based on a generalization of Bell states are quasi-perfect and, therefore, they feature the smallest error probability among all codes of the same blocklength and cardinality.