Completeness of subsystem correlations\ Consider a bipartite system AB. Suppose that many copies of the (not necessarily pure) state \\\ ho AB have been prepared. An observer Alice with access only to subsystem A can measure the expectation\ 46\ 2 Foundations I: States and Ensembles\ value of any observable of the form MAIB, while an observer Bob with access only to subsystem B can measure the expectation value of any observable of the form IA NB. Neither of these observers, working alone, can expect to gain enough information to determine the joint state \\\ ho AB.