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We consider the simplest inhomogeneous Matrix-Product-State for an open chainof N quantum spins that involves only two angles per site and two angles perbond with the following direct physical meanings. The two angles associated tothe site $k$ are the two Bloch angles that parametrize the two orthonormaleigenvectors of the reduced density matrix $\rho_k$ of the spin $k$ alone. Thetwo angles associated to the bond $(k,k+1)$ parametrize the entanglementproperties of the Schmidt decomposition across the bond $(k,k+1)$. Explicitresults are given for the reduced density matrix $\rho_{k,k+1}$ of twoconsecutive sites that is needed to evaluate the energy of two-bodyHamiltonians, and for the reduced density matrix $\rho_{k,k+r}$ of two sites atdistance $r$ that is needed to evaluate the spin-spin correlations at distance$r$. The global structure of the MPS manifold as parametrized by these $(4N-2)$angles is then characterized by its explicit Riemann metric. Finally, thegeneralizations to any tree-like structure without loops and to the chain withperiodic boundary conditions are discussed.

Explicit properties of the simplest inhomogeneous Matrix-Product-State including the Riemann metric of the MPS manifold