Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A complete picture of the four-party linear inequalities in terms of the
  0-entropy   Multipartite quantum system is complex. Characterizing the relations among
the three bipartite reduced density operators $\rho_{AB}$, $\rho_{AC}$ and
$\rho_{BC}$ of a tripartite state $\rho_{ABC}$ has been an open problem in
quantum information. One of such relations has been reduced by [Cadney et al,
LAA. 452, 153, 2014] to a conjectured inequality in terms of matrix rank,
namely $r(\rho_{AB}) \cdot r(\rho_{AC})\ge r(\rho_{BC})$ for any $\rho_{ABC}$.
It is denoted as open problem $41$ in the website "Open quantum problems-IQOQI
Vienna". We prove the inequality, and thus establish a complete picture of the
four-party linear inequalities in terms of the $0$-entropy. Our proof is based
on the construction of a novel canonical form of bipartite matrices under local
equivalence. We apply our result to the marginal problem and the extension of
inequalities in the multipartite systems, as well as the condition when the
inequality is saturated.

34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
10
The categories of this paper are: q-bio
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
11
The categories of this paper are: hep-lat
12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn