the states of the different members are distributed among a set of states $\{|\chi_i\rangle\}$, with respective frequencies $p_i$. To such situation, one assigns the density matrix $\rho_m=\sum_i p_i | \chi_i \rangle \langle \chi_i|$, which satisfies $Tr(\rho_m)=1$. It is clear that, in the $\{|\chi_i\rangle\}$ basis, the constructed matrix is diagonal, with diagonal elements given by the frequencies $p_i$. The motivation behind this definition for the density matrix comes from the fact that, in terms of $\rho_m$, the expectation values can, again, be calculated via $\langle O\rangle=Tr(\rho_m O)$. Note however that, in this case, the calculation of the expectation value depends on two types of probabilities: the quantum ones, as in the previous case, but also on the classical or *epistemic* ones, associated with a lack of knowledge of the exact state of any individual member of the ensemble. Therefore, in this case, the density matrix contains only statistical information regarding the possible state of any particular member. A closely related application of density matrices involves the description of situations in which one wants to study a closed quantum system, but one does not have full information regarding its pure state and only knows that the state is one of the members 