Density Matrix

A density operator $\rho$ is a positive, trace-one operator representing a general quantum state. A density matrix is the matrix of that operator in a chosen basis.

In Symbols

\rho\ge0, \qquad \operatorname{Tr}\rho=1, \qquad \langle A\rangle=\operatorname{Tr}(\rho A).

A pure state has

\rho=|\psi\rangle\langle\psi|, \qquad \rho^2=\rho, \qquad \operatorname{Tr}(\rho^2)=1.

A subsystem state is obtained by partial trace:

\rho_A=\operatorname{Tr}_B\rho_{AB}.

Canonical Home

Density Operators contains the definitions, state-space geometry, predictions, reduced states, examples, exercises, and references.

Context

Density operators describe pure states, randomized preparations, reduced subsystems of entangled states, thermal states, and states undergoing noise. Their matrix entries depend on basis; their eigenvalues, trace, positivity, purity, and entropy do not.

Common Aliases

density operator
density matrix
statistical operator
state operator

“Density operator” is basis independent. “Density matrix” properly refers to its array of components, although physics writing often uses the terms interchangeably.

Common Confusions

A density matrix need not be mixed; rank-one projectors are density matrices.
Off-diagonal entries do not by themselves diagnose purity because they are basis dependent.
One density operator can have many ensemble decompositions; it does not reveal one preferred hidden preparation.
A globally pure entangled state can have mixed reduced density operators.
A density operator is not a spatial mass or particle-number density.
Positivity is stronger than Hermiticity plus trace one.