they evolve as $$\label{psi_t}
    |\Psi^\pm (t)\ra=e^{-iHt} |\Psi^\pm(0)\ra.$$ By tracing out all spins except the receiver, which is taken to be the last spin $N$, one can get the density matrix of received state $$\label{rho_N_t}
    \rho_N^{\pm}(t)=Tr_{\widehat{N}} |\Psi^\pm (t)\ra \la \Psi^\pm (t)|.$$ To quantify the quality of state transfer one can compute the fidelity as $$\label{Fid1_t}
    F_u^{\pm}(t)=\la \psi^{(+ 1)}|  \rho_N^{\pm}(t) |\psi^{(+ 1)}\ra.$$ Thanks to the SU(2) symmetry of the system $F_u^{\pm}(t)$ is independent of $\theta$ and $\phi$ which means that all quantum states are transferred by the same fidelity. A general proof for this statement is given in Appendix A.

![(Color online) (a) The two fidelities $F_u^{+}(t)$ and $F_u^{-}(t)$ in terms of $Jt$ in a chain of length $N=20$ for the unrestricted measurement protocol. (b) The maximal fidelity $F_{max}^+$ as a function of $N$. (c) The maximal fidelity $F_{max}^-$ as a function of $N$. (d) The optimal time $Jt_{opt}$ versus length $N$.[]{data-label="fig2"}](Fig2.eps){width="9cm" height="7cm"}

In Fig. \[fig2\](a) the fidelity $F_u^{+}(t)$ and $F_u^{-}(t)$ are both plotted as functions of time. As it is clear from the figures the fidelity starts evolving after a certain time that information reaches the last site. Then due to constructive quantum interferences at a particular time $t=t_{opt}$ the information 