A state vector $|\psi\rangle$ describes a pure state. When preparation leaves classical uncertainty about which pure state was produced, or when part of an entangled system is ignored, the subsystem is described by a density operator $$\rho=\sum_k p_k|\psi_k\rangle\langle\psi_k|,$$ where $p_k\ge0$ and $\sum_kp_k=1$.

The density operator is Hermitian, positive and has unit trace. The expectation value of an observable $A$ is $$\langle A\rangle=\operatorname{Tr}(\rho A).$$ For a pure state, $$\rho=|\psi\rangle\langle\psi|$$ and $$\operatorname{Tr}(\rho^2)=1.$$ For a genuinely mixed state, $$\operatorname{Tr}(\rho^2)<1.$$

For an equal classical mixture of spin-up and spin-down along $z$, $$\rho=\frac12|+z\rangle\langle+z|+rac12|-z\rangle\langle-z|=rac12I.$$ Every spin direction gives equal probabilities. This same density matrix can arise as the reduced state of one particle in a maximally entangled pair, even though the combined two-particle state is pure.

If a composite state has density operator $\rho_{AB}$, the state available to subsystem $A$ is obtained by the partial trace $$\rho_A=\operatorname{Tr}B\rho{AB}.$$

Density operators unify statistical uncertainty and quantum entanglement in one formalism. They are essential for thermal quantum states, decoherence, open systems, quantum information and many-body physics.