Pure quantum state

Let $H$ be a (finite-dimensional) complex Hilbert space (see complex-hilbert-space-finite ). A pure quantum state can be specified in either of two equivalent ways:

1. State vector (ray): a unit vector $\psi \in H$ with $\|\psi\|=1$, where $\psi$ and $e^{i\theta}\psi$ represent the same physical state (global phase is unobservable).

2. Density operator (projector): the rank-one operator

   $$\rho_\psi = |\psi\rangle\langle\psi|,$$

   which is a density-operator (positive semidefinite with trace $1$).

Equivalent characterizations (finite dimension)

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

• $\rho$ is pure.
• $\rho$ has rank $1$.
• $\rho^2=\rho$ (idempotent projector).
• $\operatorname{Tr}(\rho^2)=1$ (maximal “purity” among states).
• The eigenvalues of $\rho$ are $\{1,0,\dots,0\}$.

Expectation values

If $A$ is an observable (a self-adjoint operator; see self-adjoint-operator-observable ), then in the pure state $\psi$,

Pure states are the extreme points of the convex set of all density operators; mixtures of distinct pure states produce mixed-state-quantum .