Time Evolution of ρ

States evolve. For a closed quantum system the density operator evolves by a rule that follows directly from the Schrödinger equation, and it has a clean differential form.

Unitary evolution of ρ

If a pure state evolves as $|\psi(t)\rangle = U(t)|\psi(0)\rangle$ with a unitary $U(t)$, then its density operator $\rho = |\psi\rangle\langle\psi|$ evolves as

By linearity this holds for any mixture too: each $|\psi_i\rangle\langle\psi_i|$ transforms the same way, and the probabilities $p_i$ are untouched. The "sandwich" $U\rho U^{\dagger}$ is the universal rule for closed-system evolution.

The von Neumann equation

Differentiating $\rho(t) = U\rho_0 U^{\dagger}$ with $U(t) = e^{-iHt/\hbar}$ gives the von Neumann equation (also called the quantum Liouville equation):

This is the density-operator counterpart of the Schrödinger equation. Note the sign and ordering: it is the opposite of the Heisenberg equation of motion for operators, because $\rho$ is a state, not an observable.

Properties preserved by the flow

Unitary evolution preserves everything that makes $\rho$ a valid, equally-pure state:

• Trace: $\operatorname{Tr}(U\rho U^{\dagger}) = \operatorname{Tr}(\rho) = 1$ by cyclicity.
• Hermiticity and positivity: $U\rho U^{\dagger}$ is Hermitian and positive whenever $\rho$ is.
• Purity: $\operatorname{Tr}\big((U\rho U^{\dagger})^2\big) = \operatorname{Tr}(\rho^2)$. Closed-system evolution cannot change purity, so it can never turn a pure state mixed.

That last point is important: turning a pure state into a mixed one requires either entanglement with an environment (then take a partial trace) or a non-unitary process. Pure unitary dynamics alone moves a state around the Bloch sphere without changing $|\vec r|$.

Try it

Apply a $Z$ gate to $\rho = \begin{pmatrix}3/4 & 1/4\\ 1/4 & 1/4\end{pmatrix}$ via $\rho' = Z\rho Z^{\dagger}$ (with $Z^{\dagger} = Z$), and return the coherence $\rho'_{01}$. The $Z$ gate flips the sign of the off-diagonal term, sending $1/4 \to -1/4$ — a phase flip, with the populations unchanged.

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