While a system state in quantum mechanics can be described by its state vector $\ket{\psi}$, it can also be described by a so called density operator $\rho$.

We define the density operator to be a positive operator $\rho$ with a trace equal to one ($\tr{\rho} = 1$).

This density matrix describing the state of the system for pure states is defined as $$ \rho = \sum_i p_i{\ket{\psi_i}\bra{\psi_i}} $$.