A strengthened data processing inequality for the Belavkin-Staszewski relative entropy.

In this work, we provide a strengthening of the data processing inequality for the relative entropy introduced by Belavkin and Staszewski (BS-entropy). Here, we focus on quantum channels which are conditional expectations onto subalgebras. This extends previous results by Carlen and Vershynina for the relative entropy and other standard $f$-divergences. To this end, we provide two new equivalent conditions for the equality case of the data processing inequality for the BS-entropy. Subsequently, we extend our result to a larger class of maximal $f$-divergences.