all possible input states on the surface of the Bloch sphere just as the one in Eq. (\[Fav\_ud\_t\]) to get the average fidelity $F_{av}^{odd}(t)$.

Just as before we consider the first peak of the average fidelity at the optimal time $t_{opt}$. In TABLE II we give a comparison for the average fidelity of even and odd chains versus length $N$ when the outcome of the measurement is spin up. By comparing the values one can realize that the quality of transfer is slightly lower for odd chains. For instant the average fidelity in the odd chain of length $N=19$ is $0.88$ while for a longer even chain of $N=20$ is $0.91$. This means that the SU(2) symmetry of the ground state in the even chains makes the quality of transfer even higher than the slightly shorter chains but with an odd length.

![image](Fig5.eps){width="15cm" height="5cm"}

Imperfections {#sec7}
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Preparing the system in its anti-ferromagnetic ground state needs cooling to zero temperature which in reality cannot be achieved. Hence, the initial state of the system is inevitably a thermal state of the form $\rho_{th}=\frac{e^{-\beta H}}{Z}$, where $\beta=1/K_{B}T$ in which $T$ is temperature, $K_{B}$ is the Boltzmann constant and $Z$ is the partition 