It's not in general true in quantum mechanics. It is true for 1-particle quantum mechanics if and only if the potential and any other interactions (e.g. form of canonical momentum for external magnetic field) are specified.

The reason is that the arena of quantum mechanics is not space with 3-dimensions, but configuration-space, with 3-n dimensions, one space for each particle (disregarding symmetries). Having time evolution be known lets us get rid of one spatial dimension, but we need to get rid of one spatial dimension for each of n particles. The other thing that destroys any hope is that non-local interactions are often used to model systems.

Of course quantum mechanics is only an approximation to quantum field theory, which is nicely local in the spatial sense.

Does this have anything to do with holographic theory

You know, I asked that at a colloquium nearly a year ago, and got back the answer "no", but without a satisfactory explanation.

Would it be correct to say this is still true for an uncharged black hole, based on the existence of frames of reference in which matter evaporated just before entering the event horizon?

I can't quite figure out what you're asking here, and probably couldn't give an answer without a full theory of quantum gravity.