for both $\alpha=\uparrow,\downarrow$ and thus we drop the index $\alpha$.

![(Color online) (a) The average fidelity $F_{av}(t)$ as a function of $Jt$ for a chain of length $N=20$ in a restricted basis protocol. (b) The maximal average fidelity $F_{av}(t_{opt})$ in terms of length $N$.[]{data-label="fig3"}](Fig3.eps){width="9cm" height="5cm"}

In Fig. \[fig3\](a) we plot $F_{av}(t)$ as a function of time. At $t=t_{opt}$ the average fidelity peaks for the first time. In Fig. \[fig3\](b) the maximum of average fidelity is depicted in terms of $N$ which can be well fitted by a linear function as $F_{av}(t_{opt})= -0.006 N+1.020$. This shows that for chains up to length $N\approx 60$ the average fidelity is above the classical threshold 2/3.

\[table\_I\]

          $ N $             4        6        8        10       12       14       16       18       20
  ---------------------- -------- -------- -------- -------- -------- -------- -------- -------- --------
   $F_{av}(projection)$   0.9991   0.9867   0.9735   0.9604   0.9482   0.9368   0.9264   0.9171   0.9082
   $F_{av}(attaching)$    0.9554   0.9212   0.8986   0.8826   0.8693   0.8584   0.8496   0.8425   0.8365

For the sake of completeness we compare the attainable average fidelity of our proposed mechanism for the restricted basis with the widely studied attaching procedures, in which one extra qubit that carries our desired quantum state is attached to a spin chain initialized in its 