Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The trumping relation and the structure of the bipartite entangled
  states   The majorization relation has been shown to be useful in classifying which
transformations of jointly held quantum states are possible using local
operations and classical communication. In some cases, a direct transformation
between two states is not possible, but it becomes possible in the presence of
another state (known as a catalyst); this situation is described mathematically
by the trumping relation, an extension of majorization. The structure of the
trumping relation is not nearly as well understood as that of majorization. We
give an introduction to this subject and derive some new results. Most notably,
we show that the dimension of the required catalyst is in general unbounded;
there is no integer $k$ such that it suffices to consider catalysts of
dimension $k$ or less in determining which states can be catalyzed into a given
state. We also show that almost all bipartite entangled states are potentially
useful as catalysts.

26
The categories of this paper are: bayes-an
18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
2
The categories of this paper are: math
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
17
The categories of this paper are: q-fin
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat