# How does the black hole information paradox threaten quantum determinism?

I was surprised by the subtitle of Susskind’s Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics. I would have expected that information loss would violate classical mechanics (“God does not play dice”, Einstein 1926), not quantum mechanics (“The future is unpredictable”, Feynman 1965). Susskind himself says on page 91: “If we do look [at a photon] the conservation of information fails.” I asked him about this and he replied:

“Ok, you have put your finger on an important issue that I felt was just too technical to completely spell out for the layman, but maybe I should have. In quantum mechanics the conservation of information is defined for isolated systems during the time when they are prepared and when they are observed.

“During this interval we assume the system was completely isolated from the environment which includes observers, apparatuses etc. Technically this means that the system remains in a pure state unentangled (in the technical sense) with anything else. In that case information is conserved.

“In the case of a black hole the experiment (?) consists of preparing a system of particles in some pure state—allowing it to collapse to a black hole and evaporate—and only at the end, measure the radiation. Any observation or interaction with the environment during the interval would ruin the experiment.

“To confirm that information is conserved one needs to replicate the experiment many times and observe mutually incompatible observables in different instances of the experiment. For example the system could be a particle prepared in a pure wave-packet state. It could pass through slits and hit s florescent screen. To confirm the non-loss of information would mean to detect an interference pattern but that would take many particles.

“What Hawking was saying is that even in the most ideal case of a perfectly isolated system, black holes would not be subject to the usual rules—in other words decoherence would take place without interaction with any environment.”

That leaves me with several questions, since raised with Professor Susskind, but he has not replied:

1. Why is quantum determinism, in Susskind’s view, more in need of preservation than classical determinism?

2. How is it that information loss in black holes threatens quantum determinism, but information loss in quantum mechanics does not? In his book, Susskind says “Quantum Mechanics, despite its unpredictability, nevertheless respects the conservation of information.” Isn’t this self-contradictory?

3. Hawking talks about throwing computers and encyclopaedias into black holes – classical objects losing classical information. What has that got to do with a pure quantum state decohering “without any interaction with any environment”? Is Susskind talking about virtual particles converting to the real particles of Hawking radiation? Doesn’t the gravitational pull at the event horizon count as an environment?

I'm going to try and interpret what Susskind means here.

1) Quantum determinism is encoded by the fact that unitary operations map pure states to pure states. What this means is that if you know the state of the system and the unitary operations that act on it, you know that the final state must also be pure. This is a crucial feature of quantum mechanics in that it ensures that probabilities are conserved. Mathematically, this is seen via $$\rho_{pure} = |\psi\rangle\langle{\psi}|\mapsto U|\psi\rangle\langle{\psi}|U^\dagger = |\psi'\rangle\langle{\psi'}| = \rho'_{pure}$$

The only way to go from a pure state to a mixed state is with an interaction with another system.

We ought not to care about information conservation classically since we expect classical physics to be an effective theory - the underlying proper theory being quantum.

2) There is no information loss in quantum mechanics on its own. The black hole information paradox can be seen as a violation of the equation above: your state goes from a pure state to a mixed state without interacting with an outside system. This means that the operator that took you there can't have been unitary and probabilities aren't conserved.

3) There isn't really such a thing as classical information in this context. The 'information' contained in the computer, encyclopedia etc is really encoded in its state vector, and is therefore quantum. The classical information is just an approximation to this quantum information. If you know the state of a book and the state of the black hole, then you have a pure system: $$|\psi\rangle = |\psi_{\text{black hole}}\rangle + |\psi_{\text{book}}\rangle$$ Now that you have a pure state, you can run the clock forward and backward using a unitary. In principle, if you knew the unitary that 'scrambled' your information, you could recover the information by unscrambling using the inverse of the unitary. But if your state ends up in a mixed state, then you can't: the information has been lost. The gravitational event horizon doesn't count as an environment because it must be part of the original system: you stared with some bunch of particles in a pure state, and you compacted them into a black hole which formed an event horizon. Since you didn't interact with them, the system - including event horizon, singularity etc - must form a pure state, according to quantum mechanics anyway. Quantum gravity might say otherwise.

I've been a bit colloquial in this discussion, if you want more technical details just ask.

• Could you please say just a bit more about why being compacted into a black hole wouldn't count as an interaction? My classical intuition can't conceive of anything any more violent that could happen to a bunch of particles. – D. Halsey Sep 18 '18 at 22:17
• It's an interaction, but it's one they takes place within the system. The total system remains pure, even if the subsystems that make it up don't (i.e. individual clumps of particles). Information is moved among different subsystems, but ultimately you ought to be able to reverse the entire system unitarily and recover it. – Akoben Sep 18 '18 at 22:24
• I don't read Boolean algebra but get the gist of your answer. But as I understand it, the reason we don't see dead and alive cats is because with classical objects, the decoherence time is too short to detect. So I still don't see why e.g. a book + black hole should be seen as a 'system' in a pure state. The book - and the materials it was made from - would have decohered long before it fell into the hole. Space debris is falling onto planets, stars, and black holes all the time - plenty of interaction out there. What am I missing? – Charles Whitehead Sep 23 '18 at 17:18
• The idea of a book might be a bit misleading. Really, we mean a book that we can 'prepare' in a pure quantum state (not easy in practice). Decoherence happens only by interaction with the environment, but here there is no environment: we are considering a universe only made up of a book and a black hole. We are also claiming that we know the state of the book and the black hole exactly. It is true that if we considered the book in isolation, we would observe decoherence due to the black hole, but the overall state of book + bh would remain pure. – Akoben Sep 25 '18 at 9:44