How to prove superadditivity of Holevo capacity? It is given as Exercise 2 in notes 18: \[\chi^*(\Lambda_1 \otimes \Lambda_2) \geq \chi^*(\Lambda_1) + \chi^*(\Lambda_2).\]
The trick is that since the Holevo capacity is defined as a maximum over all ensembles, in particular it’s at least as large as its value on any tensor product ensemble.
P.S. for more on the additivity of the Holevo capacity, Section 20.5 of Mark Wilde’s book has an interesting discussion.