Monogamy of Logarithmic Negativity and Logarithmic Convex-Roof Extended Negativity

Abstract: One of the fundamental traits of quantum entanglement is the restricted shareability among multipartite quantum systems, namely monogamy of entanglement, while it is well known that monogamy inequalities are always satisfied by entanglement measures with convexity. Here we present a measure of entanglement, logarithmic convex-roof extended negativity (LCREN) satisfying important characteristics of an entanglement measure, and investigate the monogamy relation for logarithmic negativity and LCREN both without convexity. We show exactly that the $\alpha$th power of logarithmic negativity, and a newly defined good measure of entanglement, LCREN, obey a class of general monogamy inequalities in multiqubit systems, $2\otimes2\otimes3$ systems and $2\otimes2\otimes2^{n}$ systems for $\alpha\geq4\ln2$. We provide a class of general polygamy inequalities of multiqubit systems in terms of logarithmic convex-roof extended negativity of assistance (LCRENoA) for $0\leq\beta\leq2$. Given that the logarithmic negativity and LCREN are not convex these results are surprising. Using the power of the logarithmic negativity and LCREN, we further establish a class of tight monogamy inequalities of multiqubit systems, $2\otimes2\otimes3$ systems and $2\otimes2\otimes2^{n}$ systems in terms of the $\alpha$th power of logarithmic negativity and LCREN for $\alpha\geq4\ln2$. We also show that the $\beta$th power of LCRENoA obeys a class of tight polygamy inequalities of multiqubit systems for $0\leq\beta\leq2$.