  Quasi-factorization-type inequalities for the relative entropy have recently
proven to be fundamental in modern proofs of modified logarithmic Sobolev
inequalities for quantum spin systems. In this paper, we show some results of
weak quasi-factorization for the Belavkin-Staszewski relative entropy, i.e.
upper bounds for the BS-entropy between two bipartite states in terms of the
sum of two conditional BS-entropies, up to some multiplicative and additive
factors.
