Reconstructing quantum states from local data

Quantum spin chains are systems of extreme complexity, in the sense that the number of parameters that fully characterize the state of a quantum spin chain grows exponentially with the number of spins. Yet, physically relevant subsets of all quantum states can be well-approximated by a small number of parameters using well-known methods such as Matrix Product States (MPS). The structure of such states can guarantee reconstruction of the state from the measurement of a small number of simple observables, merely growing linearly with the number of spins. \\ We compare two classes of quantum states which admit efficient reconstruction from incomplete, local information: States which have vanishing conditional mutual information, and the recently introduced class of states with non-decreasing operator Schmidt rank under partial traces which includes generic Matrix Product Operators (MPO). It is well-known that R\'{e}nyi entropies can be used to characterize the bond dimension of a pure MPS, i.e. the number of parameters required to describe the state. For mixed MPOs, no similar relation is known. Our comparison provides a first relation between the mutual information and the bond dimension of an MPO representation of a mixed state.