A pure state $|\psi\rangle$ determines probabilities via $p(i)=|\langle i|\psi\rangle|^2$. More generally, a density operator $\rho$ encodes mixed states and yields $p(i)=\mathrm{Tr}(\rho\,\Pi_i)$ for projective measurement elements $\Pi_i$.

A density operator satisfies $\rho\ge 0$ and $\mathrm{Tr}(\rho)=1$. A pure state obeys $\rho^2=\rho$ and $\mathrm{Tr}(\rho^2)=1$, whereas mixed states have $\mathrm{Tr}(\rho^2)<1$.

For outcome $i$, the post-measurement state becomes $\rho_i = \dfrac{M_i\rho M_i^\dagger}{\mathrm{Tr}(M_i\rho M_i^\dagger)}$. This generalizes the projection postulate and naturally describes weak or indirect measurements.

When outcomes are ignored, the state transforms as $\rho\mapsto \sum_i M_i\rho M_i^\dagger$. This map is completely positive and trace-preserving, and it often models environment-induced decoherence in practice.