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

TL;DR

This work addresses how the three bipartite reductions of a tripartite state relate, proving the rank inequality $r(\rho_{AB})\,r(\rho_{AC}) \ge r(\rho_{BC})$ and its equivalent $0$-entropy form $S_0(AB)+S_0(BC)\ge S_0(AC)$. The authors introduce a novel canonical form for bipartite matrices under local equivalence and use it to reduce the problem to a formal, inductive argument based on Schmidt rank. The main contributions include a complete proof of the four-party linear inequalities in terms of $0$-entropy, a suite of derived inequalities for the marginal problem, and an extension framework to multipartite systems along with saturation conditions. This advances understanding of marginal consistency in quantum systems and provides tools potentially applicable to broader quantum-information problems involving bipartite structures and partial transpose techniques.

Abstract

Multipartite quantum system is complex. Characterizing the relations among the three bipartite reduced density operators $ρ_{AB}$, $ρ_{AC}$ and $ρ_{BC}$ of a tripartite state $ρ_{ABC}$ 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 $r(ρ_{AB}) \cdot r(ρ_{AC})\ge r(ρ_{BC})$ for any $ρ_{ABC}$. 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.

Figures (1)

• Figure 1: The tripartite state $\rho_{ABC}$ is conjectured to satisfy the inequality $r(\rho_{AB})\cdot r(\rho_{BC}) \ge r(\rho_{AC})$ in terms of bipartite reduced density operators $\rho_{AB}$, $\rho_{BC}$, and $\rho_{AC}$, and $r(M)$ denotes the rank of matrix $M$. The inequality is known to be equivalent to the $0$-entropy inequality $S_0(AB)+S_0(BC)\ge S_0(AC)$. We prove the inequality in this paper.

Theorems & Definitions (11)

• Conjecture 1
• Definition 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Theorem 8
• Theorem 9
• Lemma 10