Density matrix

A '''density [[matrix (math)|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 matrix|density matrices]] arises because it is not possible to describe a quantum mechanical system that undergoes general [[quantum operation]]s such as [[measurement]], using exclusively states represented by [[bra-ket notation|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 [[Quantum entanglement|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 $|\backslash psi\_j\; \backslash rangle$ with probability pj, the density matrix is the sum of the projectors, weighted with the appropriate probabilities (see [[bra-ket notation]]): :$\backslash rho\; =\; \backslash sum\_j\; p\_j\; |\backslash psi\_j\; \backslash rangle\; \backslash langle\; \backslash psi\_j|$ The density matrix is used to calculate the expectation value of any operator A of the system, averaged over the different states $|\backslash psi\_j\; \backslash rangle$. This is done by taking the trace of the product of ρ and A: :$\backslash operatorname\{tr\}[\backslash rho\; A]=\backslash sum\_j\; p\_j\; \backslash langle\; \backslash psi\_j|A|\backslash psi\_j\; \backslash rangle$ 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 [[eigenvalue]]s 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 operator]]s 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 state]]s are [[extreme point]]s of the set of states. Note that using the [[GNS construction]], we can recover Hilbert spaces which realize ''A'' as an algebra of operators. [[Category:Handbook of Quantum Information]] [[Category:Mathematical Structure]] {{Template:FromWikipedia}}