 As a preliminary, von Neumann discusses the relation between states of systems and subsystems, in particular the notion of partial trace and the biorthogonal decomposition theorem (i.e. the theorem stating that an entangled quantum state can always be written in terms of perfect correlations between two special bases for the subsystems). He also shows (as mentioned above) that the usual statistics of measurements cannot be recovered by assuming that the ‘observer’ is initially in a mixed state. He then proves that it is always possible to define an interaction Hamiltonian that will correlate perfectly the eigenstates of any given observable of an ‘observed’ system with the eigenstates of some other suitable observable of an ‘observer’, leaving as an exercise for the reader to show that predictions are independent of where one places the boundary between the two.