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The contribution to the entanglement of an impurity attached to one end of a$J-J_2-delta$ quantum spin chain (S=1/2) is studied. Two different measures ofthe impurity contribution to the entanglement have been proposed: theimpurity-entanglement-entropy S_{imp} and the negativity N. The first, S_{imp},is based on a subtractive procedure where the entanglement-entropy in theabsence of the impurity is subtracted from results with the impurity present.The other, N, is the negativity of a part of the system separated from theimpurity and the impurity itself. In this paper we compare the two measures anddiscuss similarities and differences between them. In the $J-J_2-\delta$ modelit is possible to perform very precise variational calculations close to theMajumdar-Ghosh-point (J_2 = J / 2 and \delta = 0) where the system is gappedwith a two-fold degenerate dimerized ground-state. We describe in detail howsuch calculations are done and how they can be used to calculate N as well asS_{imp} for any impurity-coupling J_K. We then study the complete cross-over inthe impurity entanglement as J_K is varied between 0 and 1 close to theMajumdar-Ghosh-point. In particular we study the impurity entanglement when astaggered nearest-neighbour-interaction proportional to $\delta$ is introduced.In this case, the two-fold degeneracy of the ground-state is lifted leading toa very rapid reduction in the impurity entanglement as $\delta$ is increased.

Impurity Entanglement in the $J-J_2-\delta$ Quantum Spin Chain