Performance of polar codes for quantum and private classical communication

We analyze the practical performance of quantum polar codes, by computing rigorous bounds on block error probability and by numerically simulating them. We evaluate our bounds for quantum erasure channels with coding block lengths between 2¹⁰ and 2²⁰, and we report the results of simulations for quantum erasure channels, quantum depolarizing channels, and 'BB84' channels with coding block lengths up to N = 1024. For quantum erasure channels, we observe that high quantum data rates can be achieved for block error rates P_e ≤ 10^-4 and that somewhat lower quantum data rates can be achieved for quantum depolarizing and BB84 channels. Our results here also serve as bounds for and simulations of private classical data transmission over these channels, essentially due to Renes' duality bounds for privacy amplification and classical data transmission of complementary observables. Future work might be able to improve upon our numerical results by employing a polar coding rule other than the heuristic used here.