Exact solution for the quantum and private capacities of bosonic dephasing channels

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. Closed formulae for these capacities in bosonic systems were lacking for a key class of non-Gaussian channels, bosonic dephasing channels, which are used to model noise affecting superconducting circuits and fibre-optic communication channels. Here we provide an exact calculation of the quantum, private, two-way assisted quantum and secret-key-agreement capacities of all bosonic dephasing channels. We prove that they are equal to the relative entropy of the distribution underlying the channel with respect to the uniform distribution, solving a problem that was originally posed over a decade ago.