Bring the chapter together: ensembles, the density matrix and its rules, purity and the Bloch ball, the von Neumann equation, entropy, composite systems, the partial trace and the Schmidt decomposition. No worked example sits above the answers.

A detective works both from what is known and from what is only probable. The density matrix is the notebook that keeps both, and this chapter’s tools are how to read it.

$\hat{\rho}=\sum p_{k}\ket{\psi_{k}}\bra{\psi_{k}}$, $\langle\hat{A}\rangle=\operatorname{tr}(\hat{\rho}\hat{A})$, $i\hbar\,d\hat{\rho}/dt=[\hat{H},\hat{\rho}]$, $S=-\operatorname{tr}\hat{\rho}\ln\hat{\rho}$, $\hat{\rho}_{A}=\operatorname{tr}_{B}\hat{\rho}$.

• A density matrix is Hermitian, non-negative and of trace one.
• Isolated evolution keeps eigenvalues, purity and entropy.
• A pure whole has equally mixed parts; they are pure exactly when it is a product.