Entanglement in J_1-J_2, S=1/2 quantum spin chains with an impurity is
studied using analytic methods as well as large scale numerical density matrix
renormalization group methods. The entanglement is investigated in terms of the
von Neumann entropy, S=-Tr rho_A log rho_A, for a sub-system A of size r of the
chain. The impurity contribution to the uniform part of the entanglement
entropy, S_{imp}, is defined and analyzed in detail in both the gapless, J_2 <=
J_2^c, as well as the dimerized phase, J_2>J_2^c, of the model. This quantum
impurity model is in the universality class of the single channel Kondo model
and it is shown that in a quite universal way the presence of the impurity in
the gapless phase, J_2 <= J_2^c, gives rise to a large length scale, xi_K,
associated with the screening of the impurity, the size of the Kondo screening
cloud. The universality of Kondo physics then implies scaling of the form
S_{imp}(r/xi_K,r/R) for a system of size R. Numerical results are presented
clearly demonstrating this scaling. At the critical point, J_2^c, an analytic
Fermi liquid picture is developed and analytic results are obtained both at T=0
and T>0. In the dimerized phase an appealing picure of the entanglement is
developed in terms of a thin soliton (TS) ansatz and the notions of impurity
valence bonds (IVB) and single particle entanglement (SPE) are introduced. The
TS-ansatz permits a variational calculation of the complete entanglement in the
dimerized phase that appears to be exact in the thermodynamic limit at the
Majumdar-Ghosh point, J_2=J_1/2, and surprisingly precise even close to the
critical point J_2^c. In appendices the relation between the finite temperature
entanglement entropy, S(T), and the thermal entropy, S_{th}(T), is discussed
and and calculated at the MG-point using the TS-ansatz.