Density Matrix

A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states. (In contrast, a pure state is described by a single state vector). The density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics.

Explicitly, suppose a quantum system may be found in state with probability p₁, or it may be found in state with probability p₂, or it may be found in state with probability p₃, and so on. Then the density matrix for this system is

The expectation value for any observable A is given by

that is, the expectation value of A for the mixed state is the sum of the expectation values of A for each of the pure states, weighted by the probabilities p_i.

Mixed states arise in situations where the experimenter does not know which particular states are being manipulated. Examples include a system in thermal equilibrium (at finite temperatures) or a system with an uncertain or randomly-varying preparation history (so one does not know which pure state the system is in). Also, if a quantum system has two or more subsystems that are entangled, then each subsystem must be treated as a mixed state even if the complete system is in a pure state. The density matrix is also a crucial tool in quantum decoherence theory.

The density matrix is a representation of a linear operator called the density operator. (The close relationship between matrices and operators is a basic concept in linear algebra.) In practice, the terms "density matrix" and "density operator" are often used interchangeably. Both matrix and operator are self-adjoint (or Hermitian), positive semi-definite, of trace one, and may be infinite-dimensional. The formalism was introduced by John von Neumann (and independently but less systematically by Lev Landau and Felix Bloch in 1927).