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The inverse problem of 'eigenstates-to-Hamiltonian' is considered for an openchain of $N$ quantum spins in the context of Many-Body-Localization. We firstconstruct the simplest basis of the Hilbert space made of $2^N$ orthonormalMatrix-Product-States (MPS), that will thus automatically satisfy theentanglement area-law. We then analyze the corresponding $N$ Local Integrals ofMotions (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 theDiagonal Ensemble Density Matrix that allows to compute long-time-averagedobservables of the unitary dynamics. Explicit results are given for the memoryof local observables and for the entanglement properties in operator-space, viathe generalized notion of Schmidt decomposition for density matrices describingmixed states.

Construction of Many-Body-Localized Models where all the eigenstates are Matrix-Product-States