function. The transport mechanism is just the same as before. The projective measurement on the first qubit and the following unitary dynamics transfers information across chain just as the case that system has been initialized in its ground state. In fact, the assumption of a unitary evolution is valid only when the thermalization time is much longer than our optimal time $t_{opt}$. In Fig. \[fig5\](a), the maximal attainable fidelity $F_{max}$ is plotted in terms of $K_BT/J$ for a chain of length $N=10$. As SU(2) symmetry remains valid in the thermal initial state, the fidelity is independent of the basis of measurement. As it is evident from the figure, there is a plateau for $F_{max}$ at low temperatures which its width is determined by the finite size energy gap of the system. It is worth mentioning that the optimal time at which the fidelity peaks does not change with temperature which is consistent with the results of [@bayat-thermal].

In practical situations, it is impossible to isolate the system from its environment. To study such effects, we assume that the system is initialized in its ground state and the projective measurement is performed on the first spin just as before. However, we 