of the measurement is the singlet $|B_0\ra$. After measurement the first sites of the two chains get entangled and hence at any time $t$ the quantum state of the system can be written as $$\label{psi_ent_t}
    |\psi(t)\ra=2 e^{-iH_Tt} P^{B_0}_{1_L,1_R} |GS\ra_L \otimes |GS\ra_R,$$ where $P^{B_0}_{1_L,1_R}=|B_0\ra \la B_0|$ projects the first sites of the two chains (i.e. spins at sites $1_L$ and $1_R$ as depicted in Fig. \[fig2\](b)) into a singlet state $|B_0\ra$, the factor 2 at the beginning of the formula is for normalization and $H_T=H\otimes I+I\otimes H$ is the total Hamiltonian of the system. One can compute the reduced density matrix of the last two sites by tracing out all the rest. The special symmetries of the system and conservation of parity during the evolution implies that $$\label{psi_ent_t}
    \rho_{N_L,N_R}(t)=\frac{1}{2}\left(
                        \begin{array}{cccc}
                          a(t) & 0 & 0 & 0 \\
                          0 & 1-a(t) & b(t)& 0 \\
                          0 & b(t) & 1-a(t) & 0 \\
                          0 & 0 & 0 & a(t) \\
                        \end{array}
                      \right)$$ where both $a$ and $b$ are real numbers and can be written as

$$\begin{aligned}
 \label{a_b_t}
  a(t) = \frac{1}{2} &\{& \la\uparrow\Downarrow| e^{+iHt} |\uparrow_N\ra \la\uparrow_N|  e^{-iHt} |\uparrow\Downarrow\ra \times \la\downarrow\Uparrow| e^{+iHt} |\uparrow_N\ra \la\uparrow_N| e^{-iHt} |\downarrow\Uparrow\ra \cr
       &+& \la\uparrow\Downarrow| e^{+iHt} |\uparrow_N\ra \la\uparrow_N|  