other eigenvector corresponding to the negative eigenvalue (with eigenvalue $-1$) is $$\label{psi_m1}
    |\psi^{(-1)}\ra=\cos{(\frac{\theta}{2})}|\downarrow\ra- e^{-i\phi} \sin{(\frac{\theta}{2})}|\uparrow\ra.$$ one can transfer one of these eigenvectors to another by using the following unitary operator $$\label{R_unitary}
R_u=|\psi^{(+1)}\ra \la \psi^{(-1)}|+|\psi^{(-1)}\ra \la \psi^{(+1)}|.$$

To initialize the quantum state $|\psi^{(+1)}\ra$ in the sender site we measure the Hermitian operator $\overrightarrow{\sigma}.\overrightarrow{{\bf n}}$ at site 1. With probability of 1/2 the outcome is $+1$ and the initialization is done otherwise with probability of 1/2 the output is $-1$ and thus the unitary operator $R$ should act on site 1 to convert its state into $|\psi^{(+1)}\ra$. As the result of this measurement the quantum state of the whole system changes accordingly. Depending on the outcome of the measurement the quantum state of the system initialized to one of the following states $$\begin{aligned}
 \label{initi1_0}
  |\Psi^+(0)\ra &=& \sqrt{2} P^{(+1)} |GS\ra \cr
  |\Psi^-(0)\ra &=& \sqrt{2} R_u P^{(-1)} |GS\ra\end{aligned}$$ where $P^{(\pm 1)}=|\psi^{(\pm 1)}\ra \la\psi^{(\pm 1)}|$ are the projecting operators and $\sqrt{2}$ is the normalization factor. Each of these states are obtained by probability of 1/2 and as it is clear the unitary operation $R_u$ acts only when the outcome of the measurement is $|\psi^{(-1)}\ra$.

Since neither of these states are the eigenvector of the Hamiltonian 