pure vs mixed state

A pure state is a quantum system in a single, definite state vector — one ket |ψ⟩ that captures everything there is to know about it. Even a pure state can be uncertain about a measurement, because a superposition spreads quantum amplitude across several outcomes, but that uncertainty is purely quantum: the system genuinely has no sharper description. A mixed state, by contrast, is a probabilistic blend of several possible states, reflecting ordinary ignorance about which one the system is actually in.

The distinction is easy to confuse but important. A superposition of spin-up and spin-down is still a pure state: the system is in one definite state that happens to point sideways, and a suitable measurement can confirm it with certainty. A mixture of spin-up and spin-down is different: the system is really in one or the other, you just do not know which, and no measurement gives a sure result. The crucial difference is interference — pure superpositions can interfere, classical mixtures cannot.

The density matrix tells the two apart cleanly. A pure state's density matrix is a single projector and satisfies a simple test (squaring it leaves it unchanged); a mixed state's does not. Crucially, an entangled pair can be in a perfectly pure overall state while each partner, viewed alone, is in a mixed state. This is how the quantum uncertainty of entanglement masquerades, locally, as plain statistical ignorance — and it underlies decoherence and the emergence of classical behaviour.