interacting qubits for which the interaction type is Heisenberg with the exchange coupling $J$. A local control is available for the first qubit to operate a quantum gate or perform a spin measurement. (b) A Bell measurement on the first qubits of two noninteracting chains (note that labeling of the atoms are reversed in each chain) is used for entanglement distribution along the two spin chains. []{data-label="fig1"}](Fig1.eps){width="8cm" height="6cm"}

Introducing the model {#sec2}
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We assume a system of $N$ spin-1/2 particles interacting via an anti-ferromagnetic Heisenberg Hamiltonian $$\label{H}
    H=\sum_{k=1}^{N-1} J_k \overrightarrow{\sigma}_k.\overrightarrow{\sigma}_{k+1}$$ where $\overrightarrow{\sigma_k}=(\sigma_k^x,\sigma_k^y,\sigma_k^z)$ is the vector of Pauli operators acting on site $k$ and $J_k$ is the exchange coupling which is assumed to be uniform (i.e. $J_k=J>0$ for all $k$’s) unless it is stated. A schematic picture of this system is shown in Fig. \[fig1\](a). System is cooled down to its ground state $|GS\ra$. For the moment we consider even chains (i.e. even $N$) in which due to the SU(2) symmetry of the Hamiltonian the ground state is unique and lies in the subspace that half of the spins are up. Moreover, in even chains the SU(2) symmetry implies that the reduced density matrix of each spin is maximally mixed. 