I know that the title of the question is rather vague, so let me clarify what I mean.
For a quantum system, the set of states has infinitely (even continuously) many extreme points, i.e. there are infinitely many pure states. By the set of states, I mean the convex set of density operators $\mathcal{S} \subseteq \text{Herm}(\mathcal{H})$ on a Hilbert space $\mathcal{H}$ rather than the set of unit vectors in the Hilbert space $\mathcal{H}$. So I am talking about a convex subset of the vector space of Hermitian operators on $\mathcal{H}$ (for my puroposes, we can restrict to finite dimensions).
For example, the set of states for a spin-1/2 particle is given by the Bloch ball. Every point on the boundary of the Bloch ball is an extreme point of the Bloch ball and therefore a pure state. This boundary is a sphere, containing continuously many states.
I am wondering whether there are physicists who have come up with reasons why the set of states of a (quantum) system should be discretized instead of continuous in the sense that it should only have finitely many extreme points / pure states. To see geometrically what I mean, see the following figure:
More precisely, my question is: What are possible reasons or motivations why the set of states of a (quantum) system should be the convex hull of finitely many points (i.e. a polytope), instead of having infinitely many pure states?
Looking for such reasons, I have come across papers like this one:
Discreteness and the origin of probability in quantum mechanics
They say that quantum-gravitational considerations might suggest some kind of discretization of the states of a system. First of all, I don't really understand their argumentation yet why there should be discreteness. Moreover, I'm not sure whether the kind of discreteness they propose is the same as the one I'm asking for (e.g. I'm not sure whether their state spaces are convex).
If anyone can help me with this kind of argumentation (by explaining it or pointing to additional sources) or if anyone has come across other arguments for discreteness, I would be glad to see them.
I'm aware of the fact classical systems can be described by a simplex, therefore bearing this kind of discreteness. However, I'm interested in reasons why quantum state spaces should be discretized.
EDIT:
Thanks for the answers so far. Perhaps I should be clear about a few things: I do not think that quantum state spaces are discrete, and I do not really try to justify discreteness of quantum state spaces. I also see reasons why a quantum state space cannot be discrete. I am just curious whether other physicists have come up with justifications or reasons for discreteness. Why? To see what ideas are ruled out by arguments against discreteness.
The answers so far are not uninteresting, but they provide reasons why a quantum state space should not be discrete. I see that my question somehow asks for justification of something that cannot be, but I am still interested in what possible motivations for discreteness there could be.