Re: information conservation
If we don't know the quantum physics behind this well-known thermodynamic process
We do. Quantum statistical thermodynamics is a very well established field. It's what gives you Boe-Einstein condensation, the Chandrasakar limit etc. It doesn't violate determinism.
Heisenberg's uncertainty principle should have led us to question the general validity of determinism in any case.
No. HUP doesn't violate determinism. A least not per-se. Maybe worth pointing out that detirminism in the quantum sense is the uniqueness of the time evolution of the state vector (which is what completely describes the system), not of classical variables.