A complete picture of the four-party linear inequalities in terms of the 0-entropy

Abstract: 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.