Probability distributions consistent with a mixed state

A density matrix $\rho$ may be represented in many different ways as a mixture of pure states, $\rho = \sum_i p_i |\psi_i\ra \la \psi_i|$. This paper characterizes the class of probability distributions $(p_i)$ that may appear in such a decomposition, for a fixed density matrix $\rho$. Several illustrative applications of this result to quantum mechanics and quantum information theory are given.