The MPS (matrix product state) formalism gives a recipe to construct Hamiltonians in quantum spin chains from $n$-tuples of $k\times k$- matrices. This $n$-tuple defines a completely positive map and the existence of the uniform spectral gap of the Hamiltonian is related to the spectral property of the associated CP map. I would like to talk about a classification problem of this class of Hamiltonians. Through the relation between Hamiltonians and CP maps, the problem is reduced to the question of path connectedness of a class of CP maps.