Proof of Conservation of Information

[gabeeisenstei]

I'm having trouble with the idea of conservation of information.

I watched Susskind introduce the concept of entropy as "hidden information", using an example of drops of water filling a bathtub: a message encoded in the sequence of drops is lost for practical purposes, but in principle is recoverable at the micro-level. I don't understand this recoverability "in principle".

Let's simplify with an ideal box, into which particles are shot through a window (which is subsequently closed). Can we use momentum and the tracing of a trajectory as the encoding of relevant information? Another piece of information might simply be the time intervals between identical particles shot into identical boxes with identical momentum.

In the first case, let the velocities be the same, and the difference be the angle at which the particle goes through the window (or where it first strikes the far wall of the box). It seems to me that there are at least two different initial angles that will, after some number of bounces around the box, settle into the same trajectories. At that point the information as to their initial angles would seem to be lost.

In the second case, it seems clear that pairs of particles entering the box separated by different time intervals will attain the same trajectories if one interval is a multiple of the other, in terms of the time needed to travel across the box.

What am I missing? Is my box too ideal, not absorbing any of the particle's momentum? Is it wrong even to say that a single particle bounces around in the box, since a photon striking a mirror should actually be described as exciting an electron which then emits a different photon? Or would you apply the uncertainty principle to deny the hypothesis of identical particles with identical momentum? (But then the uncertainty principle would serve to destroy information, not preserve it?)
Where is the proof that information is never lost?

***

I wrote the above before viewing previous threads about conservation of information. I see that many people here don't believe in it or don't think it is well defined. I was assuming that it is widely acknowledged, given the prominence of the "information paradox" generated by black holes. If people like Hawking didn't assume conservation of information (at least outside of black holes), there would be no paradox. Surely he thinks there is a proof?

 

[Jano L.]

Where is the proof that information is never lost?

Whether the "information" (distinction in initial conditions) is preserved depends on the model of the evolution. There are models that preserve it, and models that violate it.