Quantum capacity and codes for the bosonic loss-dephasing channel

Bosonic qubits encoded in continuous-variable systems provide a promising alternative to two-level for quantum computation and communication. So far, photon loss has been the dominant source of errors bosonic qubits, but significant reduction recent qubit experiments suggests that dephasing should also be considered. However, detailed understanding combined channel is lacking. Here, we show that, unlike its constituent parts, loss-dephasing non-degradable, pointing towards richer structure this channel. We bounds capacity use numerical optimization find optimal single-mode codes wide range error rates.