density matrices
where the wi are positive-valued probabilities (they sum up to 1), the vectors  αi are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. We can interpret ρ as representing an ensemble where 




w

i




{\displaystyle w_{i}}

 is the proportion of the ensemble whose states are 




|


α

i


⟩


{\displaystyle |\alpha _{i}\rangle }

. 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.
