Polar Codes for Private and Quantum Communication Over Arbitrary Channels

We construct new polar coding schemes for the transmission of quantum or private classical information over arbitrary quantum channels. In the former case, our coding scheme achieves the symmetric coherent information, and in the latter, the symmetric private information. Both schemes are built from a polar coding construction capable of transmitting classical information over a quantum channel. Appropriately merging two such classical-quantum schemes, one for transmitting amplitude information and the other for transmitting phase, leads to the new private and quantum coding schemes, similar to the construction for Pauli and erasure channels of Renes et al. The encoding is entirely similar to the classical case, and thus efficient. The decoding can also be performed by successive cancellation, as in the classical case, but no efficient successive cancellation scheme is yet known for arbitrary quantum channels. An efficient code construction is unfortunately still unknown. Generally, our two coding schemes require entanglement or secret-key assistance, respectively, but we extend two known conditions under which the needed assistance rate vanishes. Finally, although our results are formulated for qubit channels, we show how the scheme can be extended to multiple qubits. This then demonstrates a near-explicit coding method for realizing one of the most striking phenomena in quantum information theory: the superactivation effect, whereby two quantum channels, which individually have zero quantum capacity can have a nonzero quantum capacity when used together.