Polar Codes for Erasure and Unital Classical-Quantum Markovian Channels
Jaswanthi Mandalapu, Vikesh Siddhu, Krishna Jagannathan
TL;DR
The paper investigates classical communication over cq-channels with Markovian memory, focusing on erasure and unital noise models. It shows that Arıkan polar codes can achieve the classical capacity for these channels when the receiver has channel-state information (and Pauli-ordering for the unital case), by combining induced classical-channel capacity results with a finite-state approximation of countable Markov processes. The authors establish a four-step proof: reduce to the induced classical channel, approximate countable Markov memory by finite FAIM channels, apply FAIM-polar results, and extend to the original countable-state model via Portmanteau. This work provides an explicit, low-complexity coding scheme showing capacity-achievability for Markovian cq-channels and highlights the broader applicability of polar codes in quantum settings with memory.
Abstract
We consider classical-quantum (cq-)channels with memory, and establish that Arıkan-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.