Density matrix

Density matrix is a mathematical representation of a quantum state that generalizes the state vector to include both pure states and mixed states — statistical ensembles of quantum states. While a state vector |ψ⟩ describes a system that is definitely in a particular superposition, a density matrix ρ can describe a system whose exact state is unknown, whether due to classical ignorance or because the system is entangled with an environment that has been traced out.

The density matrix is defined as ρ = Σᵢ pᵢ |ψᵢ⟩⟨ψᵢ|, where pᵢ are probabilities and |ψᵢ⟩ are state vectors. For a pure state, the density matrix is idempotent: ρ² = ρ. For a mixed state, this fails. The von Neumann entropy S = −Tr(ρ log ρ) quantifies the mixedness, ranging from zero for pure states to a maximum for completely mixed states. The density matrix formalism is essential in quantum entanglement and quantum statistical mechanics, where partial trace operations reveal how a subsystem appears when its correlations with the whole are ignored.