A density matrix, or density operator, is used in quantum theory to describe the statistical state of a quantum system. The formalism was introduced by John von Neumann (according to other sources independently by Lev Landau and Felix Bloch) in 1927. It is the quantum-mechanical analogue to a phase-space density (probability distribution of position and momentum) in classical statistical mechanics. The need for a statistical description via density matrices arises because it is not possible to describe a quantum mechanical system that undergoes general quantum operations such as measurement, using exclusively states represented by ket vectors. In general a system is said to be in a mixed state, except in the case the state is not reducible to a convex combination of other statistical states. In that case it is said to be in a pure state.
Typical situations in which a density matrix is needed include: a quantum system in thermal equilibrium (at finite temperatures), nonequilibrium time-evolution that starts out of a mixed equilibrium state, and entanglement between two subsystems, where each individual system must be described by a density matrix even though the complete system may be in a pure state. See quantum statistical mechanics.
The density matrix (commonly designated by ρ) is an operator acting on the Hilbert space of the system in question. For the special case of a pure state, it is given by the projection operator of this state. For a mixed state, where the system is in the quantum-mechanical state ∣ψj⟩ with probability pj, the density matrix is the sum of the projectors, weighted with the appropriate probabilities (see bra-ket notation):
ρ = ∑jpj∣ψj⟩⟨ψj∣
The density matrix is used to calculate the expectation value of any operator A of the system, averaged over the different states ∣ψj⟩. This is done by taking the trace of the product of ρ and A:
tr[ρA] = ∑jpj⟨ψj∣A∣ψj⟩
The probabilities pj are nonnegative and normalized (i.e. their sum gives one). For the density matrix, this means that ρ is a positive semidefinite hermitian operator (its eigenvalues are nonnegative) and the trace of ρ (the sum of its eigenvalues) is equal to one.
C*-algebraic formulation of density states
It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable. For this reason, observables are identified to elements of an abstract C*-algebra A (that is one without a distinguished representation as an algebra of operators) and states are positive linear functionals on A. In this formalism, pure states are extreme points of the set of states. Note that using the GNS construction, we can recover Hilbert spaces which realize A as an algebra of operators.