Density-operator state

Let $\mathcal{H}$ be a finite-dimensional Hilbert space. A density-operator state (often just “density matrix”) is an operator $\rho$ on $\mathcal{H}$ such that:

1. Positivity: $\rho \ge 0$, meaning $\langle\psi|\rho|\psi\rangle \ge 0$ for all $|\psi\rangle \in \mathcal{H}$.
2. Normalization: $\operatorname{Tr}(\rho)=1$, where $\operatorname{Tr}$ is the trace .

This is the concrete operator version of a quantum density operator .

Given an observable $A$ (a self-adjoint element of the observable algebra ), the expectation value in the state $\rho$ is

which defines the quantum expectation value .

Pure vs mixed

• $\rho$ is pure (a microstate ) iff $\rho = |\psi\rangle\langle\psi|$ for some normalized $|\psi\rangle$; see pure state .
• $\rho$ is mixed iff it is not pure; see mixed state .

A useful algebraic purity test in finite dimension is:

Physical interpretation

• $\rho$ encodes all measurement statistics for the system.
• Mixed states represent classical uncertainty about which microstate was prepared and/or entanglement with an environment.
• For a subsystem of a larger system, the appropriate state is the reduced density operator obtained by the partial trace .

Key properties

• Convexity: If $\rho_1,\rho_2$ are density operators and $0\le \lambda\le 1$, then $\rho=\lambda\rho_1+(1-\lambda)\rho_2$ is also a density operator. This formalizes “statistical mixing.”

• Spectral form: Since $\rho$ is positive and trace one, it diagonalizes as

  $$\rho = \sum_i p_i |i\rangle\langle i|, \qquad p_i \ge 0,\ \sum_i p_i = 1,$$

  so its eigenvalues act like a probability distribution over orthonormal eigenvectors.

• Entropy: The thermodynamic-like uncertainty in $\rho$ is quantified by the von Neumann entropy

  $$S(\rho) = -\operatorname{Tr}(\rho\log\rho).$$

• Thermal equilibrium: Given a Hamiltonian $H$ and inverse temperature $\beta$, the canonical equilibrium state is the quantum Gibbs state

  $$\rho_\beta = \frac{e^{-\beta H}}{\operatorname{Tr}(e^{-\beta H})},$$

  where the denominator is the quantum partition function .