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arXiv · 1701.03397

Polar Codes for Arbitrary Classical-Quantum Channels and Arbitrary cq-MACs

Rajai Nasser·Joseph M. Renes
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Abstract

We prove polarization theorems for arbitrary classical-quantum (cq) channels. The input alphabet is endowed with an arbitrary Abelian group operation and an Ar{\i}kan-style transformation is applied using this operation. It is shown that as the number of polarization steps becomes large, the synthetic cq-channels polarize to deterministic homomorphism channels which project their input to a quotient group of the input alphabet. This result is used to construct polar codes for arbitrary cq-channels and arbitrary classical-quantum multiple access channels (cq-MAC). The encoder can be implemented in $O(N\log N)$ operations, where $N$ is the blocklength of the code. A quantum successive cancellation decoder for the constructed codes is proposed. It is shown that the probability of error of this decoder decays faster than $2^{-N^{\beta}}$ for any $\beta<\frac{1}{2}$.

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Rajai Nasser, Joseph M. Renes. 2017-01-12. Polar Codes for Arbitrary Classical-Quantum Channels and Arbitrary cq-MACs. https://doi.org/10.1109/tit.2018.2869460

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