  We design quantum compression algorithms for parametric families of tensor
network states. We first establish an upper bound on the amount of memory
needed to store an arbitrary state from a given state family. The bound is
determined by the minimum cut of a suitable flow network, and is related to the
flow of information from the manifold of parameters that specify the states to
the physical systems in which the states are embodied. For given network
topology and given edge dimensions, our upper bound is tight when all edge
dimensions are powers of the same integer. When this condition is not met, the
bound is optimal up to a multiplicative factor smaller than 1.585. We then
provide a compression algorithm for general state families, and show that the
algorithm runs in polynomial time for matrix product states.
