The first one is that it is impossible to determine the entanglement of a particular quantum state. The proof is as following:

Assume that you have a detector that can detect if the state of two spins is entangled or not. So we have (here |0> is the “null” state of the detector and U is the unitary operator determining the interaction of the detector with the spins)

U |PSI0>|0> = |PSI0>|:-)>,

if the state |PSI0> of the two spins is entangled, and

U |PSI1>|0> = |PSI1>|:-(>,

if the state |PSI1> of the two spins is not entangled.

The state |+>|+> is not entangled, so we have

U |+>|+>|0> = |+>|+>|:-(>,

and also

U |->|->|0> = |->|->|:-(>.

Now consider the entangled state |+>|+>+|->|->. Because of the linearity of quantum evolution, we must have

U( |+>|+>+|->|->)|0> = ( |+>|+>+|->|->)|:-(>.

This proves that the “entanglement detector” cannot work in general.

The second problem has to do with information storage. If you say that “a single observer can see both bits, measuring the early radiation and then jumping in and seeing the copy behind the horizon,” you have to account for how the observer can store the information describing the state of the early radiation. The number of the micro-states of the early radiation is larger than exp(A/4), where A is the present area of the black-hole event horizon. The information required to describe a particular state (which is the log of the number of the states) is larger than A/4. Now, the holographic principle implies that the observer, in order to be able to store the information, has to be surrounded by an area larger than A. The observer has to be larger than the black hole! This is an alternative explanation of why the observer cannot just “jump in.”

Can someone perhaps study this exact energy eigenstate, which in the quantum sense is static, with tools not available for dynamically evolving states (which are awkward superpositions of energy eigenstates and often incredibly complicated to analyse)? I’m thinking, in particular, of the possibility of calculating the frequency spectrum of the Hawking radiation – which we should perhaps rename “the Hawking standing-wave pattern” since it’s not really radiating any more. Would this help to reveal whether the near-horizon environment is “violent” – a firewall – or “gentle” – a classical-GR-like quiet place?

Just a thought! I hope this thought experiment is useful to someone!

Given that you assume you have these brutal bits of knowledge, namely which photons are early, and which are late, and how big the black hole is and where it is, I don’t see any reason to suppose you can entangle the remaining coherence in the early radiation distant from the black hole and learn anything at all about the emissions of the late-classical black hole from the radiation. I think all you have shown is that the separation “early” and “late” is just not compatible with a unitary S-matrix for the black hole.

Just by measuring a black hole’s approximate position and approximate horizon location, you are restricting it’s thermal ensemble in a way that prevents certain kinds of entanglement from surviving. While I don’t see any proof that what I am saying is right, I also don’t see any guarantee that the implicit entanglement involves in measuring that the radiation is early leaves the early radiation state pure enough to do the measurements you need. Obviously if it does, your argument goes through, but the very fact that your have a paradox must mean it is not so— in order to determine the late radiation, you need to mae measurements on the “early” radiation over such a very long time that you aren’t even sure when you are done if it is early or late. This means that you are working over an entire black hole S-matrix event, not separating it into a two-step scattering where you know something about the intermediate black hole state (that it is a certain size, with a certain amount and kind of early radiation).

So, as far as I see, there is an additional unjustified assumption here, which is common to all the referenced literature about the Page time, namely that it is possible to simultaneously produce semiclassical black hole states entangled with pure-enough early Hawking radiation to make measurements on the whole set of early Hawking radiated particles which determine something about the late radiation. This is a heuristic assumption, and I think all you are doing is showing that it is false.

The only case in which I can see this early/late separation is completely justified is if you throw something into a highly charge black hole, and wait for the hole to decay to extremality, and look at _all_ the radiation emitted during this process (so all the radiation is “early” in this definition, since once the black hole is extremal again, it’s cold asymptotic S-matrix state). In this case, the arguement is surely completely coherent, and the end-state of the black hole is a known pure-state, it’s an extremal black hole with charge Q and velocity V (assuming a perfectly BPS model black hole, so there is no further decay). Then you can measure the outgoing radiation state, and determine the Q and V of the final state.

But in this case, the end result is no longer decaying at all, so there is no paradox, no thermal horizon and no Hawking radiation. The only time you get a paradox is when the late-state black hole is truly thermal and truly macroscopically entropic, so any intuition that associates the GR solution to a quantum state of some sort is not particularly clear (you have to associate the GR solution to a thermal ensemble).

So I can’t internalize the argument enough to see whether it is correct, it seems obviously wrong (but that’s only because the holographic complementarity seems obviously right to me), and the sticking point in understanding the argument for me is the heuristic regarding describing hugely entropic black holes using some sort of unknown quantum state for the black hole alone, rather than an entangled state of the black hole and all the radiation, early and late, with no way to make the distinction between early and late without completely ruining the ability to measure anything interesting at all about the late state.

So while I don’t find the argument persuasive, it’s only because I don’t buy the assumptions in the related literature on Page times (assumptions which don’t appear in Page’s original paper, I should add). I am questioning these obscure assumptions, not the detailed stuff in the latest paper.

In Susskind’s reply, since he at times made similar arguments about early and late radiation, he also ends up using a classical black hole picture, and pretends that you can talk about the state of infalling mater and outgoing early/late radiation separately and coherently. So Susskind already implicitly internalized this framework, and perhaps this is the reason the argument was persuasive for him. I would not give up complementarity for this, or honestly, just about anything barring someone taking an instrument and throwing it into a black hole and getting a contradiction with complementarity. It’s just too obviously correct to be false.

Regarding the “firewall” resolution, it is not satisfactory, because the firewall stress, in the same semiclassical approximation, is nonzero on the horizon, and falls inward along with anything else. This means that the singularity needs to constantly replenish the firewall with new stress by some crazy mechanism, something which is not really reasonable at early times. To see this, consider charged black holes, because the domain of communication with the singularity does not extend past the Cauchy horizon (which degenerates to r=0 in the neutral limit).

I really think that this is finding an inconsistency in the implicit assumptions in the Page time literature, not in black hole theory itself. This is very interesting and important, but please don’t discard complementarity, as I think it is almost surely fine as is.

- AMPS state that since the energy is finite, the Hawking radiation can be considered as living in a finite-dimensional Hilbert space. I don’t see how this is the case. In a theory with massless particles (e.g. photons), the initial ‘stuff’ which made the black hole could have consisted of arbitrarily many quanta (of sufficiently small energy), giving an infinite-dimensional Hilbert space. This point might not actually be important though…

- I don’t understand the relation between B and C. B is some field mode, corresponding to part of the ‘late’ Hawking radiation. C is then “…its interior partner mode.” What does that mean?

- As far as I can tell, AMPS (and Bousso in his follow-up) are taking “A and B are maximally entangled” to be equivalent to S_AB = 0. That doesn’t correspond to my understanding of entanglement. A two-qubit system in the state |00> has zero entropy, but the two qubits are not entangled. For the same reason, I don’t see how different field modes are entangled in (say) the Minkowski vacuum; all the oscillators are in their ground state.

- There is a part of the (seemingly crucial) discussion about entropy at the bottom of page 4 which I don’t understand. We have a three-part system ABC, and strong sub-additivity of entropy, S_AB + S_BC >= S_B + S_ABC. The inequality S_AB < S_A is argued for, and this seems fine (the Hawking radiation becomes 'more pure' as more of it is emitted). But then the following sentence appears: "The absence of infalling drama means that S_BC = 0 and so S_ABC = S_A." Let's take S_BC = 0 for granted; I don't see why that implies the second equality. The general inequality is S_ABC <= S_A + S_BC, with equality only if A and BC are uncorrelated, in the sense that the density matrix for the whole system is just the tensor product of those for the two subsystems, A and BC.

I’ll save you the effort of replying to this and add what Don explained to me: You’ve assumed the inside modes to be independent of the outside modes, so whatever goes on with the outside state by making the measurement at I^+ doesn’t affect the negative energy modes, so you can’t expect them to cancel. (Did I finally get this straight?) There’s two problems I have with that. First, that assumption isn’t explicitly stated. And second, I still don’t see why a reshuffling of occupation numbers is the only way to encode information in the outgoing radiation. And if that’s not what you do, how would the infalling observer notice the difference?

This is all far-fetched, to say the least. First, black holes are predicted to exist by general relativity, and by now there are some generally accepted astronomical observations of such objects out there. Second, Hawking radiation is far from being a conjecture. It is a prediction of Standard Model physics near the black hole horizon. There is no Planck-scale physics involved, just SM and classical GR. The quantum gravity effects are important only near the black hole singularity, while the horizon is quite well described with the classical theory.

Therefore, the BH information paradox is a real puzzle to solve, not just some conjectured scenario. Even without ever doing any real experiments near the horizon of any astronomical BH, we still have two theories (SM and GR) which lead to a paradoxical situation when combined to describe information loss in black holes. This is a conceptual problem with one of the two theories (or both) and needs to be resolved, regardless of any lack of experimental data. The so-far-unknown theory of quantum gravity is expected to give a resolution of the paradox, so thought-experiments on this topic are extremely useful for people doing research in quantum gravity.

The only conjecture in the whole story is whether AdS/CFT does or does not have anything to do with non-AdS quantum gravity. While personally I don’t believe AdS/CFT is applicable to the real-world gravity, it is a legal research avenue to assume otherwise, and discuss its implications to the BH information problem.

So please, folks, this isn’t some conjectures-all-around-non-realistic-theoretical-mumbo-jumbo-made-up-for-easy-paycheck problem. It is a quite real theoretical incompatibility between QM and GR, and needs to be addressed, one way or another.

HTH

@Tom: The questions you’re asking, although important, don’t really have anything to do with the issue at hand. For the current purposes, Prof. Polchinski is assuming that AdS/CFT is correct (for which there is a huge amount of evidence), and using it to provide guidance on quantum gravity more generally. You may think that is the wrong approach, but that’s fine; this is trying to push back the boundaries of knowledge, and there will be disagreements (and mistakes) along the way.

Your questions about applications of AdS/CFT to condensed matter are pertinent, even though they’re irrelevent here. For what it’s worth, I’ve heard quite a bit about this, and I’m very skeptical. What people seem to do in practice is pick a simple gravitational theory in AdS, do some fairly easy classical calculations, re-interpret the results in terms of a hypothesised dual field theory, and then go looking for a complicated condensed matter system which it might match. It doesn’t seem to offer any real understanding, in my opinion.

(Also, the fluid/gravity correspondence mentioned by Dilaton is really completely different, and purely classical.)