It is instructive to consider particular examples to see how the DFS and computational space look, i.e., for the case of $N=2$ and $N=4$ atoms.

#### **Two atoms:** 

In this case it is easy to plot the complete Hilbert space (including states outside DFS) as it consists only of 4 states as depicted in figure \[fig2\](a). The separation into DFS states and non-DFS states is easily done in the familiar singlet-triplet basis. The DFS consists of two states: the one with two atoms in the ground state and the singlet state, i.e., the antisymmetric combination of one single excited state. Thus, we can encode one logical decoherence free qubit. The other two states are superradiant, i.e., they decay with an enhanced decay rate of $2 \Gamma_\oned$.

#### **Four atoms:** 

The complete Hilbert space consists of $2^4=16$ states, and the dimension of the DFS, shown in figure \[fig2\](b), is ${ {4 \choose 2} } = 6$. As aforementioned, we want to use as computational subspace the tensor product of the antisymmetric pairs described in equation \[eq:logicalQB1\] and \[eq:logicalQB2\] , which consist only of $2^{2}=4$ states. This is why in this situation we have to distinguish within the DFS between the computational 