Example for projective geometry used in quantum mechanics In the book The Road to Reality by Roger Penrose, projective geometry as developed during the Renaissance is framed as (eventually) playing a pivotal role in quantum mechanics. (In fact, Penrose seems enamored with the idea that there is some connection between painting and physics, particularly where twistor theory is concerned. The book's epilogue is the most blatant example of this, though other examples abound.)
Not having studied quantum mechanics, I can't really imagine how projective spaces would be used to formalize what I know intuitively about the way QM works. 
Can someone provide a very simple example?
 A: 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. 
A: Wigner's theorem is a way to understand this connection, e.g.
https://arxiv.org/abs/1112.2133
https://arxiv.org/abs/0712.0997
