Adirian: You aren't reproducing lost information; you're generating new information which is equivalent to the lost information.

An interesting but ultimately futile attempt at semantic hair-splitting.

If I redirect the two beams of a polarizing splitter together, I get back the same polarization that went in - regardless of what that incoming polarization was. I am not producing a new beam from scratch that "just happens" to match the incoming polarization, because if I change the incoming polarization, the outgoing polarization that has been produced as "new information" changes in a systematically corresponding way.

By definition, by Liouville's Theorem, and by the Markov property of time, when information can be systematically "generated as new" in a way that precisely corresponds to old information, that old information has not been "lost".

Furthermore, information about incoming polarization shows up mathematically in the relative phase of the split beams, so only someone who believes the wavefunction is a hallucination would think the information has been "lost". It's right there in the amplitude distribution.