Entanglement

Entanglement is a property of a composite quantum state. The joint state cannot be factored into independent states for each subsystem, so measurements on the parts can show correlations that are not explained by assigning each part its own standalone state.

That is the important structure: the composite system carries information that is not reducible to the pieces separately.

Pure And Mixed States

A pure state is represented by a state vector, or equivalently by a rank-one density matrix. It is not merely what appears after a measurement; it is the simplest complete state description in the formalism.

A mixed state is represented by a density matrix that is not rank one. It can come from classical uncertainty over preparations, or from looking at a subsystem of a larger entangled state.

Superposition

Superposition means a state is expressed as a linear combination of basis states. The word depends on the basis being used, so it should not be treated as a separate substance added to the system.

Interference is the observable reason superposition matters. Amplitudes combine before probabilities are measured.

Correlations And Teleportation

Mutual information measures shared information. Nonlocal quantum correlations are correlations in entangled systems that violate classical expectations under the right experimental setup. Quantum teleportation uses entanglement plus classical communication to transfer an unknown quantum state, not matter or faster-than-light information.