Quantum superposition

Remarks

Every nonzero linear combination $\chi$ gives a superposition $\chi / \norm{\chi}$. If $\psi_1, \dots, \psi_k$ are orthonormal, the condition $\norm{\psi} = 1$ reads $\abs{c_1}^2 + \dots + \abs{c_k}^2 = 1$.

Example: in $\C^2$ with orthonormal basis $\lvert 0 \rangle, \lvert 1 \rangle$, the vector $\lvert + \rangle = (\lvert 0 \rangle + \lvert 1 \rangle)/\sqrt{2}$ has norm $1$, so it is a superposition of $\lvert 0 \rangle$ and $\lvert 1 \rangle$. The vector $(\lvert 0 \rangle - \lvert 1 \rangle)/\sqrt{2}$ has the same moduli of coefficients, but it is orthogonal to $\lvert + \rangle$, so it is a different state.

Non-example: the mixed state $\frac12 \lvert 0 \rangle\langle 0 \rvert + \frac12 \lvert 1 \rangle\langle 1 \rvert = \frac12 I$ is not a superposition of $\lvert 0 \rangle$ and $\lvert 1 \rangle$. It is not of the form $\lvert \psi \rangle\langle \psi \rvert$ for any unit vector $\psi$, since its square is $\frac14 I$, not itself.