Density Operator

Operator

A density operator satisfies

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

For a normalized pure state,

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

Expectation values use the trace rule:

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

For finite-dimensional states,

$$\frac{1}{d} \le \operatorname{Tr}(\rho^2) \le 1,$$

with the lower bound reached by the maximally mixed state.

Assumptions

• In finite dimensions, $\rho$ is a positive semidefinite trace-one matrix.
• In infinite dimensions, $\rho$ should be positive trace class with trace one.
• Matrix entries depend on the chosen basis.
• Different preparation ensembles can define the same density operator.

Common Mistakes

• Treating diagonal entries as probabilities without specifying the basis.
• Forgetting positivity; Hermiticity and trace one are not sufficient.
• Confusing density operators with spatial probability densities.
• Using a non-trace-class operator as a normalized state in infinite dimensions.

Canonical Links

• Density Operators
• Density Operator Symbol
• Density Matrix Expectation
• Trace-Class and Hilbert-Schmidt Operators

References

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