Probability distributions consistent with a mixed state

A density matrix ρ may be represented in many different ways as a mixture of pure states, $\rho ={\sum }_i{p}_i|{\psi }_i〉〈{\psi }_i|.$ This paper characterizes the class of probability distributions ${(p}_i)$ that may appear in such a decomposition, for a fixed density matrix ρ. Several illustrative applications of this result to quantum mechanics and quantum information theory are given.