Classical and quantum polar coding for Bosonic Gaussian channels.

We construct explicit codes for transmitting classical or quantum information over the thermal-noise Bosonic Gaussian channel. For classical information transmission, our codes achieve the classical capacity, recently characterized by Giovannetti et al. [Nature Photonics 8, 796 (2014)], while in the quantum case they transmit at a rate given by the Gaussian coherent information. The construction is based on concatenating polar codes with a suitable discretization of the channel input into a finite constellation of coherent states. The codes have efficient encoders, but the complexity of decoding is unknown. Our results are based on a quantum version of Wu and Verdu0027uu0027s technique of determining the achievable rates of finite constellations for the classical Gaussian-noise channel [Allerton 2010, pp. 620].