The concept of a weak measurement is fascinating. I haven’t looked at the paper but if it all fits nicely with Shannon that’d be mathematically very nice and suggest some interesting metaphysical consequences vis a vis universe = information.

One thing I just want to check I understand is the paragraph ending: “For the runs where both weak measurements detected nothing, the final state was exactly the original, equal superposition of 0 and 1!”

If you ever hear a cat meow then that particular run of the experiment doesn’t count. This means that there are three cases:
1) No measurements.
2) Listen once, no meow.
3) Listen once. flip the cats state, listen again, still no meow.

If I understand what you’ve written correctly then the long-run distribution over the state of the cat in case 3 is the same as in case 1. Further, in case 2 the distribution is biased towards cat-death.

Thanks,
Christo