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. For the square matrix section, see square matrix. ... Fig. ... Fig. ... John von Neumann in the 1940s. ... Lev Davidovich Landau (Ле́в Дави́дович Ланда́у) (January 22, 1908 – April 1, 1968) was a prominent Soviet physicist and winner of the Nobel Prize for Physics whose broad field of work included the theory of superconductivity and superfluidity, quantum electrodynamics, nuclear physics and particle physics. ... Felix Bloch (October 23, 1905 – September 10, 1983) was a Swiss born physicist, working mainly in the USA. Born in Zürich, Switzerland. ... A density matrix, or density operator, is used in quantum theory to describe the statistical state of a quantum system. ... In quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. ... Measurement is the determination of the size or magnitude of something. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... The term mixed state refers to a concept in physics, particularly quantum mechanics. ... A convex combination is a linear combination of data points (which can be vectors or scalars) where all coefficients are positive and sum up to 1. ... The term pure state refers to several related concepts in physics, particularly quantum mechanics and in functional analysis. ...

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. Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. ... Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. ...

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 with probability p_j, the density matrix is the sum of the projectors, weighted with the appropriate probabilities (see bra-ket notation): In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, a projection operator on a vector space is an idempotent linear transformation. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ...

The density matrix is used to calculate the expectation value of any operator A of the system, averaged over the different states . This is done by taking the trace of the product of ρ and A:

The probabilities p_j 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. On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

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. On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... C*-algebras are an important area of research in functional analysis. ... The term pure state refers to several related concepts in physics, particularly quantum mechanics and in functional analysis. ... In mathematics, an extreme point of a subset S of a real vector space is a point in S which does not lie in the open line segment joining any two points of S. Intuitively, an extreme point is a corner of S. Compare: concave, convex. ... In functional analysis, given a C*-algebra A, the GNS construction establishes a correspondence between cyclic *-representations of A and certain linear functionals on A called states. ...