in two disconnected chains the distance over which the entanglement is generated at $t=t_{opt}$ is double the distance of state transfer. In Fig. \[fig4\](b) the maximum attainable entanglement $E_{max}=E(t_{opt})$ is plotted versus distance $N$. As it is clear from the figure entanglement decays almost linearly by increasing $N$ with a small slope such that it reaches $E_{max}=0.49$ for a large distance of $N=40$.

\[table\_1\]

     $N (even)$       4        6        8        10       12       14       16       18       20
  ---------------- -------- -------- -------- -------- -------- -------- -------- -------- --------
   $F_{av}(even)$   0.9991   0.9867   0.9735   0.9604   0.9482   0.9368   0.9264   0.9171   0.9082
     $N (odd)$        3        5        7        9        11       13       15       17       19
   $F_{av}(odd)$    0.9715   0.9526   0.9367   0.9236   0.9118   0.9013   0.8915   0.8834   0.8761

Odd Chains {#sec6}
==========

So far we have only considered even chains for which the ground state is unique and supports the SU(2) symmetry with total excitation of zero. In contrast, the odd chains have doubly degenerate ground states $|GS_\uparrow\ra$ and $|GS_\downarrow\ra$ that each can be converted to another by applying $\prod_k \sigma_k^x$. In a chain of length $N$, the ground state $|GS_\uparrow\ra$ ($|GS_\downarrow\ra$) lies in the manifold of parity +1 (-1) in which $(N+1)/2$ number of spins are up (down) and the rest 