Proof of a Conjectured 0-Rényi Entropy Inequality With Applications to Multipartite Entanglement
Zhiwei Song, Lin Chen, Yize Sun, Mengyao Hu · IEEE Transactions on Information Theory · 2022
Characterizing the relations among the three bipartite reduced density operators$ \rho _{AB}$,$ \rho _{AC}$and$ \rho _{BC}$of a tripartite mixed state$ \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 extend our result to inequalities in multipartite systems, as well as the condition when the inequality is saturated.