  The capacities of noisy quantum channels capture the ultimate rates of
information transmission across quantum communication lines, and the quantum
capacity plays a key role in determining the overhead of fault-tolerant quantum
computation platforms. In the case of bosonic systems, central to many
applications, no closed formulas for these capacities were known for bosonic
dephasing channels, a key class of non-Gaussian channels modelling, e.g., noise
affecting superconducting circuits or fiber-optic communication channels. Here
we provide the first exact calculation of the quantum, private, two-way
assisted quantum, and secret-key agreement capacities of all bosonic dephasing
channels. We prove that that they are equal to the relative entropy of the
distribution underlying the channel to the uniform distribution. Our result
solves a problem that has been open for over a decade, having been posed
originally by [Jiang & Chen, Quantum and Nonlinear Optics 244, 2010].
