  Channel polarization is a phenomenon in which a particular recursive encoding
induces a set of synthesized channels from many instances of a memoryless
channel, such that a fraction of the synthesized channels becomes near perfect
for data transmission and the other fraction becomes near useless for this
task. Mahdavifar and Vardy have recently exploited this phenomenon to construct
codes that achieve the symmetric private capacity for private data transmission
over a degraded wiretap channel. In the current paper, we build on their work
and demonstrate how to construct quantum wiretap polar codes that achieve the
symmetric private capacity of a degraded quantum wiretap channel with a
classical eavesdropper. Due to the Schumacher-Westmoreland correspondence
between quantum privacy and quantum coherence, we can construct quantum polar
codes by operating these quantum wiretap polar codes in superposition, much
like Devetak's technique for demonstrating the achievability of the coherent
information rate for quantum data transmission. Our scheme achieves the
symmetric coherent information rate for quantum channels that are degradable
with a classical environment. This condition on the environment may seem
restrictive, but we show that many quantum channels satisfy this criterion,
including amplitude damping channels, photon-detected jump channels, dephasing
channels, erasure channels, and cloning channels. Our quantum polar coding
scheme has the desirable properties of being channel-adapted and symmetric
capacity-achieving along with having an efficient encoder, but we have not
demonstrated that the decoding is efficient. Also, the scheme may require
entanglement assistance, but we show that the rate of entanglement consumption
vanishes in the limit of large blocklength if the channel is degradable with
classical environment.
