in any arbitrary basis on a single spin as needed in the encoding of the previous section. Instead quantum projecting measurement may be possible only for a certain basis, let say $\sigma_z$. The outcome of the measurement is thus either $|\uparrow\ra$ or $|\downarrow\ra$ and the quantum state of the whole system collapses to $|\uparrow\Downarrow\ra$ or $|\downarrow\Uparrow\ra$ respectively. To initialize the spin into a general superposition like Eq. (\[psi\_p1\]) a further unitary operation on first site is needed. Depending on the outcome of the measurement we apply one of the following unitary operators to the first spin $$\begin{aligned}
 \label{R1_R2}
  R_\uparrow &=& |\psi^{(+1)}\ra \la \uparrow | + |\psi^{(-1)}\ra \la \downarrow | \cr
  R_\downarrow &=& |\psi^{(-1)}\ra \la \uparrow | + |\psi^{(+1)}\ra \la \downarrow |\end{aligned}$$ where $R_\uparrow$ ($R_\downarrow$) is applied if the outcome of the measurement in the $\sigma_z$ basis is $|\uparrow\ra$ ($|\downarrow\ra$) to rotate it to $|\psi^{(+1)}\ra$. The resulted states are not eigenstates of the Hamiltonian $H$ and thus system evolves accordingly. At any time $t$ one can see that the quantum state of the system is one of the following states depending on the measurement result $$\begin{aligned}
 \label{initi2_0}
  |\Psi^\uparrow(t)\ra &=&  e^{-iHt}R_\uparrow \otimes I |\uparrow\Downarrow\ra, \cr
  |\Psi^\downarrow(t)\ra &=&  e^{-iHt}R_\downarrow \otimes I |\downarrow\Uparrow\ra,\end{aligned}$$ As 