where the "w"i are positive-valued probabilities (they sum up to 1), the vectors are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. We can interpret as representing an ensemble where is the proportion of the ensemble whose states are formula_23. When a mixed state has rank 1, it therefore describes a 'pure ensemble'. When there is less than total information about the state of a quantum system we need density matrices to represent the state.