  We find a strong-converse bound on the private capacity of a quantum channel
assisted by unlimited two-way classical communication. The bound is based on
the max-relative entropy of entanglement and its proof uses a new inequality
for the sandwiched R\'{e}nyi divergences based on complex interpolation
techniques. We provide explicit examples of quantum channels where our bound
improves both the transposition bound (on the quantum capacity assisted by
classical communication) and the bound based on the squashed entanglement
introduced by Takeoka et al.. As an application we study a repeater version of
the private capacity assisted by classical communication and provide an example
of a quantum channel with negligible private repeater capacity.
