of $\{|\chi_i\rangle\}$ with respective probabilities $p_i$. In such case, one again assigns the density matrix $\rho_m=\sum_i p_i | \chi_i \rangle \langle \chi_i|$ and calculates expectation values via $\langle O\rangle=Tr(\rho_m O)$. Both of these cases are usually refereed to as *mixed* states or *proper* mixtures.

Before moving on, it is important to point out an ambiguity regarding mixed states. The problem is that, sometimes, *different* mixed states are assigned the *same* density matrix. As a result, if one is given a density matrix of a mixed state, one does not automatically know which are the possible states for the system(s) described. To illustrate the point, consider two ensembles of electrons, one with half of the electrons with spin-up and half with spin-down along $z$ and the other with half with spin-up and half with spin-down along $y$. Given that $$\label{e3}
\frac{1}{ 2}\left\{ |+ \rangle_{z} \langle+|_{z}+|- \rangle_{z} \langle-|_{z} \right\} = \frac{1}{ 2}\left\{ |+ \rangle_{y} \langle+|_{y}+|- \rangle_{y} \langle-|_{y} \right\} ,$$ it is clear that both ensembles, which clearly correspond to different physical situations, will be associated with the same density matrix.

Finally, there is the application of density matrices for the description of a subsystem of a (closed or isolated) quantum system. Suppose, 