The lattice of trumping majorization for 4D probability vectors and 2D catalysts

The transformation of an initial bipartite pure state into a target one by means of local operations and classical communication and entangled-assisted by a catalyst defines a partial order between probability vectors. This partial order, so-called trumping majorization, is based on tensor products and the majorization relation. Here, we aim to study order properties of trumping majorization. We show that the trumping majorization partial order is indeed a lattice for four dimensional probability vectors and two dimensional catalysts. In addition, we show that the subadditivity and supermodularity of the Shannon entropy on the majorization lattice are inherited by the trumping majorization lattice. Finally, we provide a suitable definition of distance for four dimensional probability vectors.