  The inverse problem of 'eigenstates-to-Hamiltonian' is considered for an open
chain of $N$ quantum spins in the context of Many-Body-Localization. We first
construct the simplest basis of the Hilbert space made of $2^N$ orthonormal
Matrix-Product-States (MPS), that will thus automatically satisfy the
entanglement area-law. We then analyze the corresponding $N$ Local Integrals of
Motions (LIOMs) that can be considered as the local building blocks of these
$2^N$ MPS, in order to construct the parent Hamiltonians that have these $2^N$
MPS as eigenstates. Finally we study the Matrix-Product-Operator form of the
Diagonal Ensemble Density Matrix that allows to compute long-time-averaged
observables of the unitary dynamics. Explicit results are given for the memory
of local observables and for the entanglement properties in operator-space, via
the generalized notion of Schmidt decomposition for density matrices describing
mixed states.
