Characterizing mixing and measurement in quantum mechanics

What fundamental constraints characterize the relationship between a mixture $\rho ={\sum }_i{p}_i{\rho }_i$ of quantum states, the states ${\rho }_i$ being mixed, and the probabilities $p_i?$ What fundamental constraints characterize the relationship between prior and posterior states in a quantum measurement? In this paper we show that there are many surprisingly strong constraints on these mixing and measurement processes that can be expressed simply in terms of the eigenvalues of the quantum states involved. These constraints capture in a succinct fashion what it means to say that a quantum measurement acquires information about the system being measured, and considerably simplify the proofs of many results about entanglement transformation.