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  • A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states. This should be contrasted with a single state vector that describes a quantum system in a pure state. The density matrix is the quantum-mechanical analogue to a phase-space probability measure 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. The density operator for this system is where need not be orthogonal and . By choosing an orthonormal basis, one may resolve the density operator into the density matrix, whose elements are The density operator can also be defined in terms of the density matrix, For an operator, the expectation value is given by In words, 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 pᵢ and can be computed as the trace of the product of the density matrix with the matrix representation of in the same basis.

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