Dear Edwin,
Sorry I mispelled your name.
I think this is what I was remembering.
Let's say that a laser/light/photons pass through an optical system. They start out as \Psi and end up as \Psi'. Let the optical system be described as
D such that, \Psi_{end}=D\Psi
[math]\Psi_{end}=D\Psi[/math]
Let the optical system consist of
[math]D = CBB^{-1}A[/math]
As you'll notice, B and B^{-1} cancel out. Somehow, light that is blocked at B^{-1} reappears after B cancels out B^{-1}.
So where is the information/light hidden between B^{-1} and B?
One of the properties of wave-functions is that they can cancel out. This is one of the reasons I think that (1) the quantum vacuum is made out of wave-functions, and (2) that space-time is emergent from the quantum vaccum. Space-time is made out of wave-functions.
In the case of redshift losing information, but gaining it back after blueshift, the information was never lost. I think the information remains within the wave-functions of the quantum vacuum, hidden until there is enough blue shift to express the information. Can loss of information due to redshift bare any similiarity to loss of focus (bluriness)?
You asked: "Also, consider a photon that's never been red-shifted, but then falls into a hole. As it is blue-shifted, then information must be being created."
A new photon is one that is emitted from an atom. That same atom probably absorbed an incoming photon. But what happens to the information carried by the incoming photon?
In my opinion, there is nebulous quantum activity that is unmeasureable and undetectable. Much of it is noise, junk, artifacts of information chunkage, and maybe an occasional hidden signal.
It would be a great experiment to try.