Monogamy of Logarithmic Negativity and Logarithmic Convex-Roof Extended Negativity

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 αth power of logarithmic negativity, and a newly defined good measure of entanglement, LCREN, obey a class of general monogamy inequalities in multiqubit systems, 2otimes2otimes3 systems and 2otimes2otimes2ⁿ systems for αgeq4ln2. We provide a class of general polygamy inequalities of multiqubit systems in terms of logarithmic convex-roof extended negativity of assistance (LCRENoA) for 0leqβleq2. 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, 2otimes2otimes3 systems and 2otimes2otimes2ⁿ systems in terms of the αth power of logarithmic negativity and LCREN for αgeq4ln2. We also show that the βth power of LCRENoA obeys a class of tight polygamy inequalities of multiqubit systems for 0leqβleq2.