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

Abstract

We prove polarization theorems for arbitrary classical-quantum (cq) channels. The input alphabet is endowed with an arbitrary Abelian group operation and an Arıkan-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 $\beta <\frac{1}{2}$.

30 pages. Submitted to IEEE Trans. Inform. Theory and in part to ISIT2017

References in corpus (6)

• Polarization for arbitrary discrete memoryless channels
• Quantum union bounds for sequential projective measurements
• Polar codes in network quantum information theory
• Ergodic Theory Meets Polarization. I: An Ergodic Theory for Binary Operations
• Ergodic Theory Meets Polarization. II: A Foundation of Polarization Theory
• Fourier Analysis of MAC Polarization

Cited by in corpus (3)

• Multivariate Fidelities
• Quantum Polarization of Qudit Channels
• Polar Codes for Quantum Reading