Projective Hilbert space

Remarks

Every ray contains unit vectors, and two unit vectors $\psi, \varphi$ lie on the same ray if and only if $\varphi = e^{i\theta}\psi$ for some $\theta \in \R$. So $\mathbf{P}(\mathcal{H})$ is also the unit sphere of $\mathcal{H}$ modulo multiplication by phases. It is the projective space of the vector space $\mathcal{H}$; for $\mathcal{H} = \C^{n+1}$ it is $\mathbb{CP}^n$.

Example: in $\C^2$, the vectors $(1, i)$ and $(i, -1) = i(1, i)$ lie on the same ray.

Non-example: $(1, 0)$ and $(0, 1)$ lie on different rays, since $(0, 1) = c(1, 0)$ would force $1 = 0$. Also $(1, 1)$ and $(1, -1)$ lie on different rays, although their entries have the same moduli: a relative phase between the components changes the ray, an overall phase does not.