same information that the original state vector $| \psi \rangle$. In fact given a density matrix satisfying $\rho_p^2=\rho_p$ and $Tr(\rho_p)=1$, one can recover the corresponding unity norm vector of the Hilbert space (up to an irrelevant complex phase). Moreover, it is easy to show that one can use $\rho_p$ to compute expectation values via $\langle O\rangle=Tr(\rho_p O)$. Note that this identification of the (mathematical notion) trace with the (physical concept) expectation value is legitimate only if one assumes the Born rule and the collapse postulate. That is, such mathematical object can be interpreted in terms of an expectation value, i.e., in terms of what one expects the average of a large number of measurements on an ensemble of identically prepared systems to be, only if one assumes that during each measurement, one finds as a result one of the eigenvalues of $O$, with probabilities given by the Born rule. This clarifications might seem trivial at this point but will be crucial latter when we evaluate the claims regarding the alleged achievements of decoherence.

Density matrices can also be used to describe ensembles of identical systems in which not all members of the ensemble are prepared in the same state. Instead, 