a well-defined state so $\rho_A$ cannot represent its state. Now, if one considers observables of $S$ of the form $O_s= O_A \otimes \mathbb{I}$, then it is easy to show that $\langle O_s\rangle=Tr(\rho_A O_A)$. Therefore, as we said above, $\rho_A$ does not contain any information regarding the actual physical state of the subsystem in question – such subsystem simply does not possess one. Instead, it contains information regarding expectation values of *measurements* that could be performed on such a subsystem. Density matrices of this type are called *reduced* density matrices or *improper* mixtures, and satisfy $Tr(\rho_A)=1$.

At this point we would like to make a few comments that are crucial in order to evaluate the claims regarding the capacity of decoherence to solve foundational problems. These comments have to do with similarities and differences regarding mixed states and reduced density matrices. Regarding similarities, it is clear that, mathematically speaking, they are identical. That is, they are both generically represented by matrices with trace equal to one. As a result, entangled subsystems and ensembles are often described with matrices of identical form. Regarding differences, it is central to keep in mind that the physical situations described by mixtures (or proper mixtures) and 