Polar Codes for Erasure and Unital Classical-Quantum Markovian Channels

TL;DR

This paper demonstrates that Arıkan's polar codes can achieve the classical capacity of certain classical-quantum channels with memory, specifically for qubit erasures and unital noise, by leveraging classical coding guarantees and properties of Markov processes.

Contribution

It extends polar coding theory to classical-quantum channels with memory, showing capacity achievement for specific noise models using classical coding techniques.

Findings

Polar codes achieve capacity for qubit erasure channels with memory.

Polar codes achieve capacity for unital qubit channels with memory.

No entanglement is needed to attain capacity in these quantum channels.

Abstract

We consider classical-quantum (cq-)channels with memory, and establish that Ar{\i}kan-constructed polar codes achieve the classical capacity for two key noise models, namely for (i) qubit erasures and (ii) unital qubit noise with channel state information at the receiver. The memory in the channel is assumed to be governed by a discrete-time, countable-state, aperiodic, irreducible, and positive recurrent Markov process. We establish this result by leveraging existing classical polar coding guarantees established for finite-state, aperiodic, and irreducible Markov processes [FAIM], alongside the recent finding that no entanglement is required to achieve the capacity of Markovian unital and erasure quantum channels when transmitting classical information. More broadly, our work illustrates that for cq-channels with memory, where an optimal coding strategy is essentially classical, polar codes can be shown to approach the capacity.