What is the motivation for assuming "Page" scrambling for Hawking radiation?

Obviously, at the semiclassical level, we want the outgoing Hawking radiation to look thermal and mixed. However, surely there are possible pure states which are scrambled enough that it looks effectively thermal when not nearly all of them are taken into account, but yet not satisfy the "Page" property. Only Susskind calls it Page, but whatever...

That if you take less than half the Hawking radiation, it's maximally entangled with the rest. Is this assumption far stronger than necessary?

It leads to bizarre conclusions like Presskill-Hayden and AMPS.


2 Answers 2


First, it's not Page in quotation marks. It's just Page, named after Don Page, a quantum gravity researcher. In particular, people talk about the Page time which is when the black hole has evaporated one-half of the initial entropy.

Second, Page's insights are not called Page's insights just by Susskind but they're called so by most of the 200+ followups of Page's (although they may be the only ones who used the new term "Page-scrambled" for scrambling whose entanglement entropy satisfies whatever Page claimed to hold)


Third, Page has outlined arguments that the radiation is very close to thermal, indeed, but because the outgoing Hawking particle is entangled with the dual partner that falls in and modifies the black hole, the whole process of radiation increases the entanglement between the Hawking radiation that is already out and the remaining black hole – or, equivalently, the early Hawking radiation and the late one, for some boundary in between them. The entanglement is very close to the maximal one when one-half of the entropy has already been emitted.

Fourth, the AMPS argument is invalid. See most of the followups


for different explanations why it's wrong. I recommend you Raju-Papadodimas in particular.

Fifth, Preskill-Hayden is right but it is not paradoxical in any way. They show that once the entanglement entropy is forced to revert the rate, in the middle of the evaporation, it becomes possible to extract genuine information about the initial state from the Hawking radiation that is already out, assuming a huge (unrealistic, just in principle, of course) accuracy.

  • $\begingroup$ Page's article is for a uniformly random quantum state. Then, it's just a quantum information theorem. But why should we assume Hawking radiation is a uniformly random quantum state, according to the Haar measure? $\endgroup$
    – Mini Zim
    Commented Dec 14, 2012 at 12:40
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    $\begingroup$ Because the black hole is actually the fastest scrambler one may have, or among them, in the same universality class. That's kind of needed for the radiation to look thermal soon. For a modern discussion, see arxiv.org/pdf/0808.2096.pdf One doesn't need really any technicalities about the rate of the scrambling. The point is that a sufficiently old black hole simply does thermalize. Its evolution is so complicated that the state after some time $t\gg R_{BH}$ is simply random, within the conservation laws constraints. It can't be a special state because the entropy could still increase. $\endgroup$ Commented Dec 14, 2012 at 14:35
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    $\begingroup$ could a very small black hole where half of its entropy been emitted emit coherent Hawking radiation? or would the scrambling property mean that the initial states that produce the coherent Hawking radiation are micro states that are macroscopically indistinguishable from micro states that lead to Hawking radiation that is nearly thermal? $\endgroup$
    – lurscher
    Commented Dec 14, 2012 at 16:35
  • $\begingroup$ @LubošMotl, comment upgraded to question: physics.stackexchange.com/q/46860/955 also improved the wording to make the question more general (replaced 'coherent' with 'distinctly non-thermal') $\endgroup$
    – lurscher
    Commented Dec 26, 2012 at 17:27

Implicit assumption: dump N qubits in, get only N qubits out as external Hawking throughout black hole lifetime. Then, expect near maximal quantum scrambling.

But in reality, if when forming black hole, dump N qubits, well, throughout black hole lifetime, still possibility of dumping more info in. If control for no further matter falling in, OK. But realize unitarity then requires much more than N qubits worth of info (after info compression a la Shannon or its quantum von Neumann analog) radiated throughout lifetime. To account for the "ancillary" inputs of no further infalling matter/info later.

So, N qubits in, much more than N qubits out. The moral of the story is, don't expect "Page" scrambling anymore. Sorry.

If modify so that the info dumped in is spread out over lifetime of black hole, ask, do we have retro effects so that part of the info radiated comes out before that info is dumped in? If you don't like retro, the conclusion again is not "Page" scrambled. Sorry.


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