Proof of a Conjectured 0-Rényi Entropy Inequality With Applications to Multipartite Entanglement

Characterizing the relations among the three bipartite reduced density operators <tex-math notation="LaTeX">ρ AB </tex-math> , <tex-math notation="LaTeX">ρ AC </tex-math> and <tex-math notation="LaTeX">ρ BC </tex-math> of a tripartite mixed state <tex-math notation="LaTeX">ρ ABC </tex-math> 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 <tex-math notation="LaTeX">r(ρ AB) · r( ρ AC)≥ r( ρ BC) </tex-math> for any <tex-math notation="LaTeX">ρ ABC </tex-math> . 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 extend our result to inequalities in multipartite systems, as well as the condition when the inequality is saturated.