A closed timelike curve is the name in General Relativity for a time machine: you go forward in time and wind up in your past, and you go around and around the loop forever.
The point is that when you get to the same point in the loop your state must be what it was the last time you were at that point in the loop.
If you have a relativistic trajectory that doesn't form a loop in time there's no reset effect.
> Wouldn't the fact that entropy must increase mean that you can never get to the exact same state as you were before?
What you're running into is the observation that near closed timelike curves quantum mechanics inevitably becomes important.
> Consider that Heisenberg's uncertainty theorem states that we cannot know precisely the position and velocity of a particle at the same time, not even in theory. Thus, they don't have precise values even in theory.
This is a popular-level understanding of QM. The first sentence is true because it includes the all important 'at the same time'. The second is... trickier.
Position and momentum are observables, not state variables (unlike classical mechanics). They are incompatible so they suffer from Heisenberg uncertainty. But that doesn't mean a state can't have a definite position or momentum.
You can construct a wavefunction with a definite position. By Heinsenberg uncertainty, if you try to measure its momentum you can get any value. Likewise you can construct a wf with a definite momentum but if you measure its position you can get any value.
Nevertheless the system has a definite wf. For these two different wfs it is perfectly fair to say the first has a definite position and the second a definite momentum.
> Then how could you ever return back to the precisely same state which never had a precise value to begin with?
If the system goes around a CTC and returns to the same wavefunction it's the same. Quantum mechanical systems have definite states. But weirdly in QM position + momentum are not state variables.
Wouldn't the fact that entropy must increase mean that you can never get to the exact same state as you were before?
Consider that Heisenberg's uncertainty theorem states that we cannot know precisely the position and velocity of a particle at the same time, not even in theory. Thus, they don't have precise values even in theory. Then how could you ever return back to the precisely same state which never had a precise value to begin with?