Density Operator

The density operator is usually written $\rho$ . The phrase “density matrix” means the matrix representation of $\rho$ in a chosen basis.

Default Meaning

A density operator represents the quantum state of a system when pure-state vectors are insufficient or inconvenient. It satisfies

\rho^\dagger=\rho, \qquad \rho\ge0, \qquad \operatorname{Tr}\rho=1.

For a normalized pure state,

\rho_\psi = \lvert\psi\rangle\langle\psi\rvert.

Probability and Expectation Rules

Expectation values are computed by the trace rule:

\langle A\rangle_\rho = \operatorname{Tr}(\rho A).

For a projective outcome with projector $P_a$ ,

P(a)=\operatorname{Tr}(\rho P_a).

Mathematical Type

In finite dimensions, $\rho$ is a positive semidefinite matrix with trace one. In infinite-dimensional settings, it should be a positive trace-class operator with trace one.

Units

The abstract density operator is dimensionless. A probability density such as $\rho(x)$ has representation-dependent units; that is a different use of the same Greek letter.

Common Variants

$\rho_A$ : density operator for subsystem $A$ .
$\rho_{AB}$ : density operator on a composite Hilbert space.
$\rho(t)$ : time-dependent density operator.
$\rho_{mn}=\langle m\vert\rho\vert n\rangle$ : matrix element in a chosen basis.

Convention Warnings

Do not confuse a density operator $\rho$ with a spatial density $\rho(x)$ .
The diagonal entries of a density matrix are probabilities only in the basis being used.
Different preparation ensembles can define the same $\rho$ .
Positivity is stronger than Hermiticity and trace one.

Canonical Links

References

L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.