Example for projective geometry used in quantum mechanics

In classical mechanics, the state of a particle on a smooth manifold $M$ is represented as a point of the cotangent bundle $T^{\ast}(M)$. In quantum mechanics, the state of a particle on a manifold $M$ is instead represented as a unit vector in the Hilbert space $H = L^2(M)$, except that two unit vectors differing in phase (that is, differing by multiplication by a scalar) are regarded as the same state because the results of all possible experiments performed on the two states are the same. So the state of a particle is represented, not really by a unit vector in $H$, but by a point in the projective space $\mathbb{P}(H)$. Symmetries of a quantum system are then identified with continuous homomorphisms $G \to \text{PGL}(H)$ of topological groups, or with projective representations of a (probably Lie) group $G$.

In quantum computing, we might consider smaller Hilbert spaces. For example, the Hilbert space $H = \mathbb{C}^2 = \text{span}(\langle 0|, \langle 1|)$ describes a qubit, and $\mathbb{P}(H)$ is the Riemann sphere, which in this context is known as the Bloch sphere.