High order correlations of generic pure states of finite-dimensional quantum systems are determined by lower order correlations

We show that almost every pure state of multi-party quantum systems (each of whose local Hilbert space has the same dimension) is completely determined by the state's reduced density matrices of a fraction of the parties; this fraction is less than about two-thirds of the parties for states of large numbers of parties. In other words once the reduced states of this fraction of the parties have been specified, there is no further freedom in the state.