Mixed quantum state

A mixed quantum state on a finite-dimensional complex Hilbert space $H$ is a density-operator $\rho$ that is not pure (see pure-state-quantum ). Concretely, $\rho$ is mixed iff it cannot be written as $|\psi\rangle\langle\psi|$ for any unit vector $\psi$.

Equivalent characterizations (finite dimension)

For a density operator $\rho$, the following are equivalent:

• $\rho$ is mixed.
• $\rho$ has rank $\ge 2$.
• $\rho^2 \ne \rho$.
• $\operatorname{Tr}(\rho^2) < 1$.
• The von-neumann-entropy satisfies $S(\rho) > 0$.

Convex mixture form

Every density operator admits a decomposition as a convex combination of pure states:

This decomposition is generally not unique. A distinguished choice is the spectral decomposition, where the $|\psi_i\rangle$ are orthonormal eigenvectors and the $p_i$ are eigenvalues.

How mixed states arise

Mixed states appear in (at least) two mathematically distinct ways:

• Classical uncertainty (statistical mixture): the system is prepared in pure state $|\psi_i\rangle$ with probability $p_i$.
• Reduced state of a larger system: if $\rho_{AB}$ is a state on $H_A\otimes H_B$, then the subsystem state $\rho_A = \operatorname{Tr}_B(\rho_{AB})$ obtained by the partial-trace can be mixed even when $\rho_{AB}$ is pure (entanglement).

Special cases

• Maximally mixed state: on $d=\dim H$, $\rho_* = \frac{I}{d},$ which has the largest von Neumann entropy $S(\rho_*)=\log d$.