# Density matrix

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In quantum mechanics, a density matrix is a self-adjoint (or Hermitian) positive-semidefinite matrix (possibly infinite dimensional) of trace one, that describes the statistical state of a quantum system. The formalism was introduced by John von Neumann[citation needed] (and according to other sources, independently by Lev Landau[citation needed] and Felix Bloch[citation needed]) in 1927[citation needed].

The density matrix is especially helpful for dealing with mixed states, which consist of an statistical ensemble of two or more different quantum systems. (The opposite of a mixed state is a "pure state", any state with a state vector, also called ket.) The density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics.

Situations where mixed states arise, and therefore the density matrix is commonly used, include the following: A quantum system with an uncertain or randomly-varying preparation history (so one does not know with 100% certainty which pure quantum state the system is in); a quantum system in thermal equilibrium (at finite temperatures); a quantum system that started in a mixed equilibrium state and then underwent some other process. Also, if a quantum system has two or more subsystems that are entangled, then each individual subsystem must be treated as a mixed state even if the complete system is in a pure state; the density matrix of the subsystem is calculated as a partial trace of the density matrix of the whole system. Relatedly, the density matrix is a crucial tool in quantum decoherence theory. See also quantum statistical mechanics.

The operator that is represented by the density matrix is called the density operator. (The close relationship between matrices and operators is a basic concept in linear algebra; see the article Linear operator for details.) In practice, the terms "density matrix" and "density operator" are often used interchangeably. The density operator, like the density matrix, is positive-semidefinite, self-adjoint, and has trace one.