Density Operator - Knowlpedia

Let $H$ be a complex Hilbert space. A density operator (also called a density matrix) is a positive trace-class operator $\rho:H\to H$ such that:

Positivity: $\rho \ge 0$ , meaning $\langle \psi,\rho\psi\rangle \ge 0$ for all $\psi\in H$ .
Unit trace: $\operatorname{Tr}(\rho)=1$ , where $\operatorname{Tr}$ is the operator trace (Trace Operator).

In finite dimension, trace-class is automatic, so these conditions say that $\rho$ is a positive semidefinite matrix of trace one.

Basic structural facts (finite dimension)

$\rho$ is automatically self-adjoint: $\rho=\rho^\ast$ .
All eigenvalues of $\rho$ are real and lie in $[0,1]$ .
The eigenvalues sum to $1$ : if $\rho$ has eigenvalues $(p_i)$ , then $\sum_i p_i=1$ .

Thus $\rho$ admits a spectral decomposition

\rho = \sum_i p_i\,|\phi_i\rangle\langle \phi_i|,

with $(\phi_i)$ orthonormal and $p_i\ge 0$ , $\sum_i p_i=1$ .

Pure vs mixed

$\rho$ is a pure state iff it has rank $1$ , equivalently iff $\rho^2=\rho$ , equivalently iff $\operatorname{Tr}(\rho^2)=1$ . (See Pure State Quantum.)
Otherwise $\rho$ is mixed and can be written as a possibly countable mixture (a finite convex combination in finite dimension) $\rho=\sum_k q_k |\psi_k\rangle\langle\psi_k|$ with $q_k\ge 0$ , $\sum_k q_k=1$ . The sum converges in trace norm. (See Mixed State Quantum.)

Expectation values (Born rule in operator form)

If $A$ is a bounded observable (a self-adjoint operator, see Self Adjoint Operator Observable), then the expectation value in state $\rho$ is

\mathbb{E}_\rho[A] = \operatorname{Tr}(\rho A).

This formula unifies pure and mixed states. For an unbounded observable, its expectation requires additional spectral-integrability and domain conditions.

Dynamics and transformations (finite dimension)

Unitary evolution: if $U$ is unitary, then $\rho$ evolves as $\rho \mapsto U\rho U^\ast$ .
Projective measurement: spectral projectors $P_i$ (as in Spectrum Self Adjoint Finite) define outcome probabilities $\Pr(i)=\operatorname{Tr}(\rho P_i)$ .

Information-theoretic quantities

Two common functionals of $\rho$ are:

Von Neumann entropy: $S(\rho) = -\operatorname{Tr}(\rho\log\rho)$ (see Von Neumann Entropy).
Relative entropy: $D(\rho\|\sigma)=\operatorname{Tr}(\rho(\log\rho-\log\sigma))$ under suitable support conditions (see Quantum Relative Entropy).

In infinite dimension these entropy quantities can be $+\infty$ ; the logarithmic expressions require their spectral interpretation, not an assumption that every displayed product is trace-class.