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Abstract
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.
Highlights
Majorization is nowadays a well-established and powerful mathematical tool to compare probability vectors, which is applied in many and different fields as in economy, biology, physics among others
An interesting problem where majorization naturally arises is the interconversion of bipartite pure states by means of local operations and classical communication (LOCC)
In the general case, trumping majorization gives a partial order for probability vectors whose components are sorted in nonincreasing order
Summary
Majorization is nowadays a well-established and powerful mathematical tool to compare probability vectors, which is applied in many and different fields as in economy, biology, physics among others (see e.g. ref.[1] for an introduction to the topic). An interesting problem where majorization naturally arises is the interconversion of bipartite pure states by means of local operations and classical communication (LOCC). This problem consists in two parties, Alice and Bob, that share an (initial) entangled pure-state, say x Their goal is to transform x into another entangled pure-state (target state), say y , by applying LOCC. A celebrated result of Nielsen gives the necessary and sufficient condition that makes this entanglement transformation process possible[15] This process can be achieved if and only if there exists a majorization relation between the initial and target states, that is, x → y. Some results have been provided by restricting the dimension of input vectors and/or the dimension of the catalyst (see e.g. refs[27,28,29,30])
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