Quantum gain of coding by polar codes with finite codeword length

The polar codes proposed by Arikan have attracted much attention, because a channel capacity is achieved by the codes in the limit of the infinite codeword length by using successive cancellation decoding. Recently, a similar result was shown in the classical-quantum channel. In this paper, we calculate mutual information using optimum quantum decoding when the codeword length of the polar codes is finite and show that the quantum gain of coding is obtained using the polar codes. Finally, we compare the performance of the polar codes with that of other codes.