Distributional comparison for non-commutative infinitely divisible probability measures

We determine ``cumulant-type'' upper bounds of the non-commutative Wasserstein distance for certain classes of distributions $\mu$ and $\nu$, which are infinite divisible with respect to the Boolean, classical and free convolutions. The main contribution of the manuscript is an estimation of the non-commutative Wasserstein distance between $\mu$ and $\nu$, expressed in terms of the difference between cumulants of order less than $2m+4$.