Quantum Polarization of Qudit Channels

TL;DR

This paper extends quantum polar codes to qudit channels, utilizing unitary 2-designs for channel combining, and demonstrates polarization and decoding methods for Pauli qudit channels, advancing quantum communication techniques.

Contribution

It generalizes quantum polar codes to qudit channels using unitary 2-designs, including smaller subsets, and provides decoding strategies for Pauli qudit channels.

Findings

Polarization occurs for qudit channels using 2-designs.

Channel combining can be achieved with smaller Clifford subsets.

Decoding methods are developed for Pauli qudit channels.

Abstract

We provide a generalization of quantum polar codes to quantum channels with qudit-input, achieving the symmetric coherent information of the channel. Our scheme relies on a channel combining and splitting construction, where a two-qudit unitary randomly chosen from a unitary 2-design is used to combine two instances of a qudit-input channel. The inputs to the synthesized bad channels are frozen by sharing EPR pairs between the sender and the receiver, so our scheme is entanglement assisted. Using the fact that the generalized two-qudit Clifford group forms a unitary 2-design, we conclude that the channel combining operation can be chosen from this set. Moreover, we show that polarization also happens for a much smaller subset of two-qudit Cliffords, which is not a unitary 2-design. Finally, we show how to decode the proposed quantum polar codes on Pauli qudit channels.