 Solution

If the preparation label $i$ is sampled with probability $p_i$ but is not supplied to the experimenter, the quantum state ensemble is represented byFor normalized states, this density operator is Hermitian, positive semidefinite and has trace one. The states in an ensemble need not be orthogonal or distinct. For any measurement in quantum mechanics effect $E$, averaging the Born rule over the preparation label givesThus the density operator summarizes all measurements performed on the prepared system alone. For an ensemble of mixed states ${\rho }_i$, the same reasoning gives $\rho ={\sum }_i{p}_i{\rho }_i$.