Quantum state

Remarks

The unit vector $\psi$ of a pure state is unique up to a factor $e^{i\theta}$ with $\theta \in \R$. So the pure states correspond one to one to the points of the projective Hilbert space of $\mathcal{H}$, and one often calls $\psi$ itself the state, or the state vector.

A density operator $\rho$ is pure if and only if $\rho^2 = \rho$, if and only if $\operatorname{tr}(\rho^2) = 1$.

Example: on $\C^2$ with $\psi = (1, 1)/\sqrt{2}$, the state is $\rho = \frac12 \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$. It is positive with trace $1$, and $\rho^2 = \rho$, so it is a pure state.

Non-example: the identity $I$ on $\C^2$ is positive, but its trace is $2$, so it is not a state. Its multiple $\frac12 I$ is a state, but not a pure one, since $(\frac12 I)^2 = \frac14 I \ne \frac12 I$.