Uncertainty Principle state space of a single particle/object?

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.