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How you surely can't recover information from a black hole

Lots of papers that make it to the arXiv these days (but maybe a minority) are completely wrong. Some of the most crazily wrong papers that make it through the arXiv filters are those that are soon hyped by the blogs and the media. That's also the case of the preprint

How to Recover a Qubit That Has Fallen Into a Black Hole
by Aidan Chatwin-Davies, Adam S. Jermyn, and Sean M. Carroll which has been "promoted" by a guest blog written by the first co-author on the blog of the third co-author. Holy cow. This short paper is just so incredibly wrong in such an incredibly stupid way!

These individuals claim that when a quantum bit – stored in the up/down information about the spin of a spin-1/2 particle – falls into the black hole, an observer outside the black hole may recover the information quickly. In fact, these people claim that the observer may recover it as soon as the first spin-1 particle of Hawking radiation is emitted by the black hole.

They just measure the spin of the outgoing particle and the spin of the remaining black hole – and that's enough to find the qubit of a particle that has just fell beneath the event horizon, Carroll et al. claim. Wow. If one could get the information from a black hole this immediately, there would be nothing holy or black about the black hole. The object wouldn't be a black hole but a white parrot with perfect mirrors that immediately reflects and repeats everything you tell him. A black hole is just the opposite: it makes all the information as inaccessible as you can get. It's the "safest prison" or "safebox" allowed by the laws of physics.

When a particle is absorbed by a black hole, the black hole's angular momentum changes according to the usual rules of addition of the angular momentum. When the black hole emits the first Hawking particle, the rules for the addition of the angular momentum hold, too. But the first emitted Hawking particle has nothing whatever to do with the latest particle that fell into the black hole. It contains 0.0000001% of the information from the latest absorbed particle, 0.0000001% of the information from the previous one, and so on. You may wait for many additional Hawking particles but you need many of them to get a whole particular qubit.

So when the state \(\ket{K_0,J_0,M_0}_{BH}\) of a black hole decays to a superposition of tensor products \[

\sum_{K_1,J_1,M_1,k,j,m} \!\!\!\!\! c_{K_1,J_1,M_1,k,j,m} \ket{K_1,J_1,M_1}_{BH} \otimes \ket{k,j,m}

\] where \(K,k\) stands for the other quantum numbers unrelated to the angular momentum and the second tensor factor refers to the Hawking particle, then the dependence on the \(z\)-components of the spin is given by the Clebsch-Gordan coefficients or the Wigner-Eckart theorem – rules that depend on the group theory governing the addition of angular momentum. But there is no room left for any dependence that would remember anything about the most recent particle absorbed by the black hole.

The information about the infalling particle gets mixed with all the other information that the black hole carries. It means that when one Hawking particle gets emitted, you only get a tiny portion of the "soup" that contains all the previously accumulated information.

If you want to get the actual individual pieces (e.g. qubits) of information, you need to wait for very many Hawking particles. In fact, it's easy to see that you must wait up to the so-called Page time – it's the time when the entropy of the black hole has dropped (thanks to the Hawking radiation) to 1/2 of the value that the entropy had when the qubit (waiting to be recovered) was absorbed by the black hole.

Why is it so? It's because the Hawking radiation and the remaining black hole are nicely entangled at the Page time: there is a one-to-one correspondence between the possible states of the (already emitted) Hawking radiation; and the microstates of the (remaining) black hole. That's why an (insanely fine and unrealistic in practice) measurement of the information carried by the (already emitted) Hawking radiation at the Page time tells you everything about the information inside the (remaining) black hole as well – there is a one-to-one map between these two Hilbert spaces at that moment.

In \(d=4\) and using the Planck units, the Hawking evaporation takes about \(M^3\) Planck times. The Page time is comparable to 50% of this evaporation time (you may calculate the percentage exactly, it is not 50%). It is an extremely long time scale, much longer than trillions of years for a large black hole. But if you wait for a much shorter time than the Page time, you can simply not recover any particular qubits that were previously absorbed by the black hole.

Everyone in the field knows that. This paper is nothing else than the black hole researchers' counterpart of a paper claiming that \(2+2=5\). It's not not the case and everyone with the most rudimentary understanding of the black hole information must know that this paper is complete garbage.

You can "determine" the spin of the particle that fell in – if you remember that bit of information in the first place (because you copied it etc.). And yes, Carroll et al. measure the black hole even before the absorption, they remember all the changes, so what a shock that they may determine the change of the angular momentum, too (see also "ancilla bits" they keep around). But then you can't really say that the information was devoured by the black hole, the measurement of this black hole is completely useless or redundant, and claims about the recovering are vacuous.

Also, these people are confused about the very concept of a measurement – even outside the black hole context. They think that a measurement recovers the complex amplitudes in a superposition. But a measurement never does – and the complex amplitudes (in the wave function) are not measurable. Technically, they are not observables. A measurement always produces Boolean or "classical" information as the answer to a particular question about observables (all observables are represented by linear operators). When a measurement takes place, a "qubit" is transformed into a regular "classical bit".

There are other embarrassing mistakes in the paper that are unrelated to the black hole information puzzle. For example, they say that almost all the Hawking radiation from real-world black holes comes in the form of photons. However, about 1/2 of the Hawking particles (and 1/2 of the energy) are gravitons which they totally neglect.

Sociology of junk co-authored by students

Sean Carroll is clearly the "senior" co-author. Even though he has been producing nothing else than similar 100% crackpottery for a decade or so, he continues to exist in the system because the system has become pretty much completely corrupt and this jerk has contacts to the media – starting with his wife – and other powerful institutions that allow him to pretend that the garbage he writes has something to do with modern scientific research even though it doesn't.

But what about the students, Aidan Chatwin-Davies and Adam S. Jermyn? Do they have a sufficient number of friends to make the system work for them, too?

I feel some compassion because these young people have already been turned into the "folks who wrote the garbage with Sean Carroll" and it will be very hard for them to recover their neutral credentials. However, the silver lining is that our compassion has its limits because the evaluation of these young folks is really true. If these young people were any good, they would see that this claim is self-evident garbage and they would refuse to be co-authors of this garbage.

So while I am upset that individuals like Sean Carroll are able to manipulate with the system, I don't believe that the system is already corrupt enough for the likes of Chatwin-Davies and Jermyn to get a postdoc job in this discipline of physics.

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