Uncertainty Principle state space of a single particle/object? My question is about mathematics in the context of physics
Imagine a series of quantum states of a single particle:
In the first state the particle has a certain position but an uncertain momentum
In the last state the particle has an uncertain position but a certain momentum
Also imagine all other possible (obeying the Uncertainty principle) states in between
What is the mathematical space that describes all those possible states (if we disregard all other information about the particle)?
Is this space applicable only to Quantum Mechanics or to any other physical system with a similar tradeoff? By "similar tradeoffs" I mean cases like this:

a narrow spread of possible outcomes for one experiment necessarily implies a wide spread of possible outcomes for another

https://en.wikipedia.org/wiki/Quantum_state
P.S.: A similar question(s)
What is the single particle Hilbert space? is a similar question, but it talks not only about position and momentum but about all other information too
But I am only interested in the former
Also I've seen this "Uncertainty Principle for a Totally Localized Particle" post
 A: 
Imagine a series of quantum states of a single particle:
In the first state the particle has a certain position but an uncertain momentum
In the last state the particle has an uncertain position but a certain momentum
Also imagine all other possible (obeying the Uncertainty principle) states in between
What is the mathematical space that describes all those possible states (if we disregard all other information about the particle)?

States aren't uniquely determined by their position and momentum uncertainties, so there isn't a unique mathematical description of the series of states you are describing.
However, as you have alluded to, quantum states can be described as elements of Hilbert space even if there isn't a unique way to mathematically move between such states.
As for this quote

a narrow spread of possible outcomes for one experiment necessarily implies a wide spread of possible outcomes for another

I think more needs to be said. I would instead say

a sufficiently narrow spread of possible outcomes for one experiment necessarily implies a wide spread of possible outcomes for another

If the quantum state is right up against an uncertainty principle limit, then yes, narrowing the uncertainty of one observable would necessarily mean that the uncertainty of another observable would have to widen. However, if a state is not near this limit, then narrowing one uncertainty would not need to be accompanied with the widening of another. This further points to the idea that one cannot determine a unique description of the series of states that you describe.
