when you burn a piece of paper, the atoms react into new molecules, some energy is given/taken, but fundamentally you could ~~run~~ simulate the reaction in reverse and see writing in the smoke of burnt paper.

Isn't this hampered by the old 'needing a universe-sized supercomputer to run a fully accurate 1:1 simulation of the universe'? It seems like the complexity of this kind of reverse simulation would be enormous and grow bigger and more complex when you're dealing with something on the scale of black holes.

At least unless you can somehow reduce the data complexity dramatically without tainting the simulation, but that seems difficult to do when you're trying to work backwards to an unknown state. At least the pre-burnt piece of paper is known to exist alongside various rules of physics that we know - unlike in the black hole, where those rules all come with a "warrantry maybe void" asterisk.

The question isn't whether this is possible in practice or not. It's that the laws of physics are defined as differential equations of a state (think e.g. Newton's laws) so they should be "the same" forwards and backwards in time.

I thought that was only true for classical mechanics and not true for QM. Like not only are there time-dependent equations, Heisenberg says we can’t know both position and velocity of a particle which would make it hard to capture state. Also everything is expressed in terms of probabilistic events which adds to the likelihood that it’s again impossible to capture state.

The wave equation of QM is just a differential equation too. The interpretation of what its values actually are is quite contentious AFAIU, and what the Born rule used to derive the probabilities "means" depends on the interpretation.

But yes, if you have "collapsed the wave function", you don't have the information to "backtrack" anymore. But I don't think the information paradox is about this (although unsure why). AFAIU one problem is that black holes throw out even the statistical information available in the wavefunction.

I would say it's generally understood that the quantum information being radiated from a black hole is still effectively unrecoverable by any practical machine, but there's plenty of such information in our real world already. You don't even need to "go quantum" to get that result, it's not hard to have perfectly classical cases where information is completely impractically recoverable, even in thought experiments in which one can rearrange the entire physical universe to the sole task of recovering some bit of classical information, let alone any physically-realizable machine. (You can even quantify the rate of information loss in such systems with the right theories.)

What really bothers the quantum physicists is not a macro-scale unrecoverability, but the event of the loss itself. The equations of quantum physics do not have anywhere to "lose" information. If a black hole loses information, then that means there is some box you can draw in spacetime where the information goes in and doesn't come back out, and that necessarily implies that something in that box broke the fundamental equations of quantum mechanics.

I think you could analogize this issue into classical physics fairly well as finding a violation of the conservation of energy in a putatively classical system. (I have to qualify this because at it stands at the moment, conservation of energy is not absolute; at cosmological scales it seems to be violated by the expansion of the universe. So ignore that for the moment.) If someone produced a true perpetual motion machine, let alone one that actually provided over-unity energy, that would be a problem for classical physics because there is nowhere in the mathematics that energy can be produced in excess. It is written into the most fundamental formulations of the theories and backed by Noether's Theorem. Such a machine would mean that somewhere the laws of physics are being violated, and that would mean we have a hole in them. The quantum theorists are "worried" about quantum information loss in the same way a classical physicist is "worried" about over-unity machines. A real, physical quantum physicist or classical physicist might in fact be over the moon to discover such a thing; it would be a career maker and who knows what else might be implied by such a discovery. But the anthropomorphized theory that was just "violated" would be very "upset".

The other major relevant different being that the classical physics has those classic physics "in hand", in the sense that many real experiments on classic physics can be and are realized in the real world, and at this point, physicists as a whole do not take the idea that an over-unity machine is going to be produced very seriously. However, it is quite difficult to have black hole in hand to experiment on; such mathematical analogs as we have are intrinsically limited and probably quite likely to fail to exhibit the phenomena we're interested in. So there's a lot more "room" for problems to come from. Plus, classical physics already has very firmly built into it the understanding that it is only an approximation. Quantum physics is still struggling with that question; sure, we basically know it is incomplete, but we don't know exactly how to resolve that, so unlike classical physics where we know that technically, yes, it's predictions can be violated in certain ways at certain scales and we accept that, we're a lot more nervous about QM. Maybe the real theory of the universe would allow information destruction. In English, that sounds pretty harmless ultimately, like, I mean, OK, maybe that's a bit of an issue but who cares? But pushed through the aforementioned Noether's theorem it gets much more interesting considered from the mathematical point of view, and much more challenging.

re: computational complexity, I am not informed enough to answer that, but I've wondered it too. But we've said things like that before about, say, factoring huge numbers, only to find out that quantum computers can solve it with a fraction of the energy ("orders of magnitude" doesn't even cut it here).

The other thing I've wondered is how could you have a detector for those particles without interfering with your measurement (heisenberg's)? Your measurement apparatus will give off heat or very slightly change the shape of spacetime, thus interfering where your particles are detected :(.

I have no idea what I'm talking about, but quantum physics are bizarre enough that I'm not sure you can play it in reverse.

The smallest perturbations in a chaotic system yield completely different results. The slit experinment for instance suggests that measurement changes the system.

I don't understand why you would be skeptical. How do you think cell phones work?

What does “hawking radiation encoding the future through black holes” at all have to do with “humans intervening with and encoding their own signals over radio spectrums”?

Nothing.

Understood. Sorry. I should have given you the benefit of the doubt.

To me, the resolution of the paradox — that the radiated field might contain some encoded information in it — is less weird than the alternative, which is that the information somehow disappears. Information being encoded in a field seems natural to me. There’s a tight interplay between energy, temperature, and information. It’s even right there in the first law of thermodynamics: dE = TdS + dW. (The S is the missing information about the system.)

Information encoded in radiation is how we get pretty much all of the information we have about the universe. Light and radio are radiation.

But I think you're overcomplicating it. Think of it as a snapshot of an object as it crosses the event horizon. That snapshot is etched into the "surface" of the horizon and emits a pattern of radiation that corresponds to it.

It's pretty much the same as dropping a rock onto a planet and observing the crater from orbit. You left a mark on the surface and it emits a corresponding pattern of radiation.

The real crux of the matter is that we're starting to understand that information is a lot like energy. It's generally conserved in the same way, but black holes appear to destroy matter, energy, and information in a way that doesn't really add up.

So Hawking came up with the idea of Hawking radiation. It's a hypothetical way for a black hole to very slowly give back all of the matter and energy it's consumed in the form of particle radiation. If information is a conserved quantity like energy, Hawking radiation must also return all information consumed by the black hole.

Information theory is pretty interesting if you're the type of person who likes to think about black holes and cosmology in their free time.

AFAIK we haven't proven that Hawking radiation actually exists. It seems to work mathematically, but we'd have to go out and observe a black hole up close. It's also extraordinarily slow. As in, it would take the entire lifespan of the universe for a big black hole to evaporate into nothing. Ten to the power of hundreds of years. Just a hair shy of infinity.