I hadn't realized that "differential" entropy and shannon entropy are actually different and incompatible, huh.
So the case I mentioned, where you know all the positions and momentums has 0 shannon entropy and -Inf differential entropy. And a typical distribution will instead have Inf shannon entropy and finite differential entropy.
Wikipedia has some pretty interesting discussion about Differential Entropy vs Limiting density of Points, but I can't claim to understand it and whether it could bridge the gap here.
Quantum mechanics solves the issue of the continuity of the state space. However, as you probably know, in quantum mechanics all the positions and momentums cannot simultaneously have definite values.
So the case I mentioned, where you know all the positions and momentums has 0 shannon entropy and -Inf differential entropy. And a typical distribution will instead have Inf shannon entropy and finite differential entropy.
Wikipedia has some pretty interesting discussion about Differential Entropy vs Limiting density of Points, but I can't claim to understand it and whether it could bridge the gap here.