Error in an argument regarding “improper” mixtures

t wt |φt 〉〈φt | represents a system which, with probability wj , is “really in” one or another of the pure states |φj 〉. In quantum mechanics, mixtures are thought to arise in two ways — as a direct mixing of distinct pure-state preparations, and “improperly” (d’Espagnat’s term), as the reduction of a pure-state composite system to the state of a component system by tracing out the other component systems. An Argument has appeared many times in the past half-century to the effect that an attempt to apply an ignorance interpretation to an “improper” mixture leads to mathematical inconsistency. Specifically: The state of a composite system, assumed pure, is reconstructed from the purported ignorance-interpretation subensembles arising from the trace-reductions, and it is claimed that this reconstruction leads to a mixed, rather than pure, state. It is the purpose of this note to show that this Argument is incorrect. (It is not the purpose of this paper to argue further regarding the possibility of such ignorance interpretations.)