This allows us to write the the ground state $|GS\ra$ in a very generic form of $$\label{gs}
    |GS\ra=\frac{|\uparrow_k,\Downarrow\ra-|\downarrow_k,\Uparrow\ra}{\sqrt{2}},$$ where $\uparrow_k$ ($\downarrow_k$) means site $k$ is in spin up (spin down) and $\Uparrow$ represents a quantum state for the rest of the system in which there are $N/2$ spins up and $N/2-1$ spins down (similarly for $\Downarrow$ there are $N/2$ spins down and $N/2-1$ spins up). The detailed structure of $|\Uparrow\ra$ and $|\Downarrow\ra$ are very complex and due to their different parities these two states are orthogonal. In addition due to the the SU(2) symmetry of the system the generic form of the ground state in Eq. (\[gs\]) remains valid for any basis of spins.

By measuring a single spin at site $k$, in an arbitrary basis, the quantum state of the whole system collapses according to the outcome of the measurement. For instance, if the measurement is in the $\sigma^z$ basis on site $k$ then with probability of 1/2 the outcome of measurement is spin up and the quantum state of the system collapses to $|\uparrow_k,\Downarrow\ra$. This new state still remains in the subspace of the ground state but is no longer an eigenvector of the Hamiltonian and as the 