  We provide a versatile upper bound on the number of maximally entangled
qubits, or private bits, shared by two parties via a generic adaptive
communication protocol over a quantum network when the use of classical
communication is not restricted. Although our result follows the idea of Azuma
et al. [Nat. Comm. 7, 13523 (2016)] of splitting the network into two parts,
our approach relaxes their strong restriction, consisting of the use of a
single entanglement measure in the quantification of the maximum amount of
entanglement generated by the channels. In particular, in our bound the measure
can be chosen on a channel-by-channel basis, in order to make it as tight as
possible. This enables us to apply the relative entropy of entanglement, which
often gives a state-of-the-art upper bound, on every Choi-simulable channel in
the network, even when the other channels do not satisfy this property. We also
develop tools to compute, or bound, the max-relative entropy of entanglement
for channels that are invariant under phase rotations. In particular, we
present an analytical formula for the max-relative entropy of entanglement of
the qubit amplitude damping channel.
