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

TL;DR

This paper proves a key inequality relating ranks of reduced density matrices in tripartite quantum systems, providing a complete characterization of four-party linear inequalities in terms of 0-entropy, with applications to quantum marginal problems.

Contribution

It establishes the conjectured rank inequality for tripartite states, completing the understanding of four-party linear inequalities in 0-entropy terms.

Findings

Proved the inequality $r( ho_{AB}) imes r( ho_{AC}) \u2265 r( ho_{BC})$ for all tripartite states.

Constructed a novel canonical form of bipartite matrices under local equivalence.

Applied the result to quantum marginal problems and inequality saturation conditions.

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.