  We construct an explicit quantum coding scheme which achieves a communication
rate not less than the coherent information when used to transmit quantum
information over a noisy quantum channel. For Pauli and erasure channels we
also present efficient encoding and decoding algorithms for this communication
scheme based on polar codes (essentially linear in the blocklength), but which
do not require the sender and receiver to share any entanglement before the
protocol begins. Due to the existence of degeneracies in the involved
error-correcting codes it is indeed possible that the rate of the scheme
exceeds the coherent information. We provide a simple criterion which indicates
such performance. Finally we discuss how the scheme can be used for secret key
distillation as well as private channel coding.
