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It is interesting to notice that ρΩ (Ω=A,B) is block diagonal in the spin basis, each block being labeled by a spin configuration i at the boundary. Block number i is a card EiΩ ×card EiΩ matrix and has only one nonzero eigenvalue pi, the corresponding eigenvector being | R KiΩ □. In the special case where all the energies EΩ (ω,i) vanish, each state | RK iΩ □ is an equal-amplitude superposition of spin configurations. Then, each block of ρΩ is particularly simple since all its matrix elements are identical. The block labeled by the boundary configuration i, of size card EiΩ ×card EiΩ, has all its entries equal to pi /card EiΩ and a trace equal to pi.
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