On the limit relation for the quantum relative entropy

Recently, Di\'osi et al. (Int. J. Quant. Inf. 4, 99 (2006)) introduced a simple, yet very interesting model for reservoirs, in order to study the relationship between thermodynamic entropy production of a system and the corresponding von Neumann entropy (or informatic entropy, as it was called by the authors). They came up with a conjecture about the asymptotic behaviour of the state introduced in the model. The conjecture was proven later by Csisz\'ar et al. (J. Math. Phys. 48, 092102 (2007)). In this paper we give a different interpretation of the original question and extend the result to a collection of more general states.