The Senior Common Room was unusually quiet.
Professor Quillibrace was reading the newspaper.
Miss Elowen Stray was looking thoughtfully out of the window.
Mr Blottisham entered carrying a cup of tea and the expression of a man who had just discovered a serious problem with the universe.
BLOTTISHAM: I have it.
QUILLIBRACE: That is generally when I become concerned.
BLOTTISHAM: The black-hole information paradox is really quite simple.
STRAY: Is it?
BLOTTISHAM: Yes. Information goes into the black hole and gets lost.
QUILLIBRACE: You have managed to solve the paradox by repeating its name.
BLOTTISHAM: No, no. I'm being precise.
QUILLIBRACE: Then let us test the precision. What do you mean by "lost"?
BLOTTISHAM: Gone.
QUILLIBRACE: Excellent. And what do you mean by "gone"?
BLOTTISHAM: Not there.
QUILLIBRACE: And what do you mean by "not there"?
Blottisham paused.
BLOTTISHAM: I see where this is going.
QUILLIBRACE: Then we are making progress.
Elowen smiled.
STRAY: Perhaps "lost" is doing rather more work than we usually notice.
BLOTTISHAM: I don't see the difficulty. If I lose my keys, I don't have my keys.
QUILLIBRACE: Precisely. But have the keys ceased to exist?
BLOTTISHAM: No.
QUILLIBRACE: Then you have distinguished between losing something and destroying it.
BLOTTISHAM: Obviously.
QUILLIBRACE: Good. We shall now apply that obvious distinction to information.
Blottisham looked suspicious.
BLOTTISHAM: Information isn't keys.
QUILLIBRACE: An excellent beginning.
The lost keys
STRAY: Suppose you lose your keys. There are several possibilities.
BLOTTISHAM: I can't find them.
STRAY: Yes. But they might be under the sofa.
BLOTTISHAM: They might.
STRAY: Or in your other coat.
BLOTTISHAM: Also possible.
STRAY: Or you might have left them at the college.
BLOTTISHAM: Less pleasant, but possible.
STRAY: In each case, your access to the keys has changed. The keys themselves have not necessarily changed in the relevant way.
QUILLIBRACE: We therefore have a useful distinction.
He put down the newspaper.
QUILLIBRACE: Inaccessibility is not destruction.
BLOTTISHAM: Fine.
QUILLIBRACE: It becomes considerably less fine when people say, "The information is lost."
BLOTTISHAM: Because they might merely mean that nobody can get at it.
QUILLIBRACE: Correct.
STRAY: And in the black-hole problem, that distinction is not merely linguistic. It is physically decisive.
BLOTTISHAM: But if the information is behind the event horizon, then from outside we can't get it.
QUILLIBRACE: Certainly.
BLOTTISHAM: So it's lost.
QUILLIBRACE: To you.
BLOTTISHAM: Isn't that what lost means?
QUILLIBRACE: In ordinary life, often. In fundamental physics, we need rather more discrimination.
STRAY: Otherwise we slide from "unavailable from this perspective" to "no longer present in the physical state."
BLOTTISHAM: Which aren't the same.
QUILLIBRACE: Congratulations.
Hidden, scrambled, destroyed
Elowen rose and went to the blackboard.
She wrote:
ACCESSIBLE
INACCESSIBLE
SCRAMBLED
TRANSFORMED
DESTROYED
STRAY: These are not synonyms.
BLOTTISHAM: They all sound rather bad.
STRAY: For the person trying to recover the information, perhaps.
She pointed to the first word.
STRAY: Information can be accessible.
Then the second.
STRAY: It can be present but inaccessible.
Then the third.
STRAY: It can be scrambled among an enormous number of physical degrees of freedom.
Then the fourth.
STRAY: It can be transformed into a different physical organisation.
And finally—
STRAY: It could, in principle, be fundamentally destroyed.
QUILLIBRACE: The paradox concerns the last possibility.
BLOTTISHAM: So if it is merely scrambled, there's no paradox?
QUILLIBRACE: Not in the same sense.
BLOTTISHAM: Then black holes just scramble things.
QUILLIBRACE: That would be a rather premature conclusion.
BLOTTISHAM: I thought you said scrambling was fine.
QUILLIBRACE: I said scrambling is not destruction.
BLOTTISHAM: Ah.
QUILLIBRACE: You have just demonstrated why distinctions are useful.
Elowen laughed softly.
STRAY: There is something rather revealing here. The ordinary word lost compresses all these possibilities into one event.
QUILLIBRACE: Yes.
STRAY: I cannot find it.
I cannot access it.
I cannot reconstruct it.
It has been radically reorganised.
It no longer exists in the physical description.
Ordinary speech permits us to move among these without noticing.
QUILLIBRACE: Physics does not have that luxury.
The black hole as scrambler
BLOTTISHAM: So perhaps the black hole is just an extraordinarily good filing system.
QUILLIBRACE: No.
BLOTTISHAM: A bad filing system?
QUILLIBRACE: Worse.
STRAY: Perhaps a filing system that has shredded the filing system.
BLOTTISHAM: That's not particularly reassuring.
STRAY: It isn't meant to be.
QUILLIBRACE: The point is that a black hole is thought to be an extraordinarily efficient scrambler.
BLOTTISHAM: Meaning?
QUILLIBRACE: Suppose a complicated quantum state falls into it. The information associated with that state need not remain in any recognisable form.
BLOTTISHAM: So the black hole destroys the pattern.
QUILLIBRACE: No. It destroys the recognisable organisation of the pattern.
BLOTTISHAM: That sounds suspiciously like destruction.
STRAY: Not necessarily. Consider a book.
BLOTTISHAM: A book?
STRAY: Tear it into pieces.
BLOTTISHAM: I can see where this is going.
STRAY: If the pieces remain available and their relationships are known, the book can be reconstructed.
BLOTTISHAM: Yes.
STRAY: Scatter the pieces across a warehouse.
BLOTTISHAM: More difficult.
STRAY: Mix them with pieces from a million other books.
BLOTTISHAM: Practically impossible.
STRAY: But impossibility in practice is not automatically destruction in principle.
QUILLIBRACE: Precisely.
BLOTTISHAM: And a black hole does this on a rather larger scale.
QUILLIBRACE: Conceptually, yes.
BLOTTISHAM: So it is less like a cosmic shredder and more like a cosmic scrambled egg.
QUILLIBRACE: I regret saying anything about eggs.
The thermal problem
STRAY: But now we encounter the real difficulty.
BLOTTISHAM: The radiation.
STRAY: Yes.
QUILLIBRACE: Hawking radiation has a thermal character.
BLOTTISHAM: Which means it has no information.
QUILLIBRACE: No.
BLOTTISHAM: I thought thermal meant random.
QUILLIBRACE: Thermal behaviour does not automatically mean informational emptiness.
STRAY: The important question is what level of description we are using.
BLOTTISHAM: Explain.
STRAY: Imagine two decks of cards.
BLOTTISHAM: Again with the household objects.
STRAY: Each deck contains exactly the same cards.
BLOTTISHAM: Obviously.
STRAY: But the ordering can be completely different.
BLOTTISHAM: Yes.
STRAY: If I inspect only one card at a time, I learn almost nothing about the ordering of the complete deck.
BLOTTISHAM: Right.
STRAY: The information can reside in the relationships among the cards.
QUILLIBRACE: Likewise, radiation can have thermal-looking individual statistics while the complete quantum state contains highly nontrivial correlations.
BLOTTISHAM: So the information isn't necessarily in the particles.
QUILLIBRACE: Not in the sense you appear to want.
BLOTTISHAM: Where is it, then?
QUILLIBRACE: In the correlations.
BLOTTISHAM: That's not somewhere.
QUILLIBRACE: Exactly.
There was a short silence.
STRAY: That may be one of the important conceptual points.
BLOTTISHAM: Information doesn't have to have a location.
QUILLIBRACE: At least not an ordinary spatial location.
STRAY: A relation is not a thing sitting in one place.
BLOTTISHAM: So asking "Where is the information?" may already be the wrong question.
QUILLIBRACE: It may be.
BLOTTISHAM: That's becoming an irritatingly common conclusion.
QUILLIBRACE: Reality is under no obligation to avoid irritating conclusions.
The whole and the parts
STRAY: Quantum entanglement makes this even stranger.
BLOTTISHAM: I knew you were going to say that.
STRAY: The individual parts of an entangled system can appear highly random.
BLOTTISHAM: While the whole is structured.
STRAY: Exactly.
QUILLIBRACE: The correlations are not reducible to what you find by examining each component independently.
BLOTTISHAM: So you can look at every particle separately and still miss the information in the whole system.
QUILLIBRACE: Yes.
BLOTTISHAM: That's rather unfair.
QUILLIBRACE: It is quantum mechanics.
STRAY: And it gives us an important conceptual possibility for black-hole evaporation.
BLOTTISHAM: That the Hawking radiation can look thermal while the complete radiation state contains information.
STRAY: Precisely.
BLOTTISHAM: Then the information hasn't escaped as a message.
QUILLIBRACE: Correct.
BLOTTISHAM: It has become a pattern in the relationships among the radiation.
QUILLIBRACE: Much better.
BLOTTISHAM: I'm learning.
QUILLIBRACE: Slowly.
Transport or transformation?
STRAY: There is another metaphor we may need to reconsider.
BLOTTISHAM: Let me guess. Information travels.
STRAY: Yes.
QUILLIBRACE: We tend to imagine information as cargo.
BLOTTISHAM: It goes in.
QUILLIBRACE: It waits.
BLOTTISHAM: It comes out.
QUILLIBRACE: A surprisingly persistent picture.
STRAY: But perhaps the more useful idea is transformation.
BLOTTISHAM: Rather than transport?
STRAY: Yes.
QUILLIBRACE: The physical state evolves. Its correlations change. Its organisation changes. What was once represented one way may later be represented another way.
BLOTTISHAM: So nothing has to travel.
QUILLIBRACE: Not in the metaphorical sense of a little parcel called information moving through spacetime.
STRAY: The physical system changes.
BLOTTISHAM: And the distinctions can survive that change.
QUILLIBRACE: If evolution is unitary.
BLOTTISHAM: Even if the final state looks completely different?
QUILLIBRACE: Indeed.
STRAY: Preservation does not necessarily mean persistence of form.
Elowen paused.
STRAY: That may be worth keeping.
The broken glass
BLOTTISHAM: But surely irreversible things happen.
QUILLIBRACE: Of course.
BLOTTISHAM: A glass falls and smashes.
QUILLIBRACE: Yes.
BLOTTISHAM: It doesn't spontaneously reassemble.
QUILLIBRACE: Not in any realistic circumstance.
BLOTTISHAM: So the information is lost.
QUILLIBRACE: Not necessarily.
BLOTTISHAM: Again?
QUILLIBRACE: The microscopic state after the glass breaks contains vastly more detail than the macroscopic description "broken glass."
STRAY: The fragments, air, vibrations, heat and surrounding environment all participate in the resulting state.
BLOTTISHAM: So the information about the original glass has been dispersed.
QUILLIBRACE: Precisely.
BLOTTISHAM: But recovering it would be absurdly difficult.
QUILLIBRACE: Yes.
BLOTTISHAM: So it is effectively lost.
QUILLIBRACE: For practical purposes.
BLOTTISHAM: But not fundamentally.
QUILLIBRACE: That distinction is the whole point.
STRAY: The universe can make something effectively inaccessible without making it nonexistent.
BLOTTISHAM: Which means "lost" can describe our relationship to a state rather than the fate of the state itself.
Quillibrace looked up.
QUILLIBRACE: Yes.
BLOTTISHAM: I wish you would occasionally admit that I have said something useful.
QUILLIBRACE: I have just done so.
BLOTTISHAM: I hadn't noticed.
The black hole evaporates
There was another pause.
Then Elowen turned back to the board.
She erased everything except one phrase:
THE BLACK HOLE DISAPPEARS
STRAY: But now the analogy with the broken glass reaches its limit.
BLOTTISHAM: Because the black hole evaporates.
STRAY: Yes.
QUILLIBRACE: If the black hole were eternal, one could at least imagine the information remaining associated with the interior indefinitely.
BLOTTISHAM: Inaccessible, but still there.
QUILLIBRACE: Perhaps.
STRAY: But if the black hole completely evaporates, the supposed container is gone.
BLOTTISHAM: Which leaves us with the obvious question.
QUILLIBRACE: "Where did the information go?"
BLOTTISHAM: Exactly.
QUILLIBRACE: And we have just spent an hour explaining why that may be the wrong question.
BLOTTISHAM: Then what is the right one?
STRAY: Perhaps:
What physical structure now carries the distinctions that were present in the initial state?
Blottisham considered this.
BLOTTISHAM: That's considerably less catchy.
QUILLIBRACE: It is also considerably less misleading.
Information changes its address
BLOTTISHAM: So we shouldn't say the information was inside and then moved outside.
QUILLIBRACE: Correct.
BLOTTISHAM: We should say the correlations were reorganised as the system evolved.
QUILLIBRACE: Better.
STRAY: The important question becomes what structure currently carries the relevant distinctions.
BLOTTISHAM: Which means information doesn't have an address in the ordinary sense.
QUILLIBRACE: Precisely.
BLOTTISHAM: But surely something physical has to carry it.
QUILLIBRACE: Yes.
BLOTTISHAM: So information is not unreal.
QUILLIBRACE: Certainly not.
STRAY: But neither is it necessarily a substance being transported.
BLOTTISHAM: It's structure.
STRAY: Or, more carefully, we use the word information to describe structures of distinctions and correlations realised in physical states.
BLOTTISHAM: That's rather less mystical than "information is everywhere."
QUILLIBRACE: It is also considerably more defensible.
The Page curve
Blottisham sat down.
BLOTTISHAM: All right. But how do we know any of this isn't just clever wordplay?
QUILLIBRACE: An excellent objection.
BLOTTISHAM: Thank you.
QUILLIBRACE: You should enjoy it while it lasts.
STRAY: The point is that the mathematics gives us constraints.
BLOTTISHAM: The Page curve?
QUILLIBRACE: Among other things.
STRAY: Don Page considered what the entanglement between a black hole and its radiation should do if evaporation were unitary.
BLOTTISHAM: And?
QUILLIBRACE: Roughly speaking, at first the radiation becomes increasingly entangled with the remaining black hole.
STRAY: But after the Page time, the pattern should turn around.
BLOTTISHAM: Meaning?
STRAY: The later radiation must begin to carry information about the earlier state if the complete evaporation is unitary.
BLOTTISHAM: So the information eventually comes back.
QUILLIBRACE: Be careful.
BLOTTISHAM: What now?
QUILLIBRACE: You have reverted to the transport metaphor.
BLOTTISHAM: Right.
STRAY: The more precise statement is that the correlations in the radiation must evolve in such a way that the final radiation state preserves the distinctions encoded in the initial state.
BLOTTISHAM: So the Page curve isn't a story about messages escaping.
QUILLIBRACE: No.
STRAY: It is a statement about the changing organisation of quantum correlations.
BLOTTISHAM: Which means the mathematics itself pushes us away from the metaphor.
Quillibrace nodded.
QUILLIBRACE: Now you are paying attention.
Two histories
STRAY: There is an even deeper way of putting the problem.
She wrote:
A
B
STRAY: Suppose two genuinely different quantum states collapse into black holes.
BLOTTISHAM: Different initial states.
STRAY: Yes.
QUILLIBRACE: If both black holes evaporate completely into precisely the same final state, what has happened?
BLOTTISHAM: The difference has disappeared.
QUILLIBRACE: Exactly.
BLOTTISHAM: Not merely that nobody can find it.
QUILLIBRACE: No.
BLOTTISHAM: The final physical state itself no longer distinguishes A from B.
STRAY: And that is much stronger.
BLOTTISHAM: So the deepest problem isn't really that information has been misplaced.
QUILLIBRACE: Correct.
BLOTTISHAM: It's that a distinction that existed between two possible histories has ceased to exist.
STRAY: Yes.
She looked at Quillibrace.
STRAY: Which makes the paradox begin to look like a problem about identity.
QUILLIBRACE: Or, perhaps more cautiously, about the persistence of difference through physical evolution.
BLOTTISHAM: What makes A remain different from B?
QUILLIBRACE: Precisely.
BLOTTISHAM: And if physical evolution can erase that distinction, then something very fundamental has changed.
QUILLIBRACE: That is the difficulty.
The cost of genuine loss
BLOTTISHAM: But perhaps information really can be destroyed.
QUILLIBRACE: Perhaps.
BLOTTISHAM: You can't simply assume unitarity because you like it.
QUILLIBRACE: Quite right.
STRAY: The paradox would have no force if the conclusion were built into the premises.
BLOTTISHAM: So if information really is destroyed—
QUILLIBRACE: Then quantum evolution is not unitary in this situation.
BLOTTISHAM: Which would mean—
QUILLIBRACE: That a foundational principle of quantum theory would require modification.
STRAY: And that would be a very expensive theoretical conclusion.
BLOTTISHAM: Expensive?
QUILLIBRACE: Conceptually.
BLOTTISHAM: Ah.
STRAY: The strength of the paradox comes from the fact that quantum theory has survived an extraordinary number of experimental tests.
QUILLIBRACE: So abandoning unitarity is possible in principle.
BLOTTISHAM: But not something one should do merely because the metaphor of "lost information" sounds convincing.
QUILLIBRACE: Precisely.
What does the word "lost" conceal?
Elowen returned to the blackboard.
She wrote:
LOST
Then underneath:
TO WHOM?
Blottisham frowned.
BLOTTISHAM: That's interesting.
STRAY: Is information lost to an observer?
BLOTTISHAM: Or lost from the complete physical state?
QUILLIBRACE: Those are different questions.
STRAY: And there is another.
She wrote:
UNDER WHAT PARTITION?
BLOTTISHAM: Meaning?
STRAY: Suppose we divide the world into systems.
Black hole.
Radiation.
Inside.
Outside.
Observer.
Observed.
QUILLIBRACE: Then we ask where the information is located within those divisions.
STRAY: But quantum correlations need not respect our preferred partitions.
BLOTTISHAM: So the information may be in the relation between the things we've decided to separate.
QUILLIBRACE: Yes.
BLOTTISHAM: Which means our question about location may depend on how we divided the system in the first place.
STRAY: Exactly.
She paused.
STRAY: Perhaps the most interesting question is not:
Where is the information?
but:
In what relations is the information encoded?
Quillibrace nodded slowly.
QUILLIBRACE: That is considerably better.
A small rebellion against nouns
BLOTTISHAM: I notice something.
QUILLIBRACE: This should be interesting.
BLOTTISHAM: We keep replacing nouns with verbs.
STRAY: Go on.
BLOTTISHAM: Information doesn't sit somewhere.
It becomes encoded.
It becomes scrambled.
It becomes distributed.
It becomes correlated.
It becomes transformed.
QUILLIBRACE: An astute observation.
BLOTTISHAM: And the black hole isn't really a thing with information inside it.
It's a region in which physical relationships evolve in a particular way.
STRAY: Yes.
BLOTTISHAM: So perhaps our problem comes partly from turning processes into objects.
Quillibrace looked almost pleased.
QUILLIBRACE: That is a useful suspicion.
BLOTTISHAM: Thank you.
QUILLIBRACE: Do not become ambitious.
From loss to transformation
The fire had burned low.
Elowen looked again at the words on the board.
LOST
HIDDEN
SCRAMBLED
TRANSFORMED
STRAY: Perhaps we have now changed the question.
BLOTTISHAM: From "Where did the information go?"
STRAY: To:
What happened to the informational structure of the state?
QUILLIBRACE: Better.
BLOTTISHAM: And if the answer is that the structure was transformed into correlations in the radiation—
QUILLIBRACE: Then the information has not been fundamentally destroyed.
BLOTTISHAM: Even if nobody can practically reconstruct it.
QUILLIBRACE: Correct.
STRAY: And if the final state truly contains no distinction corresponding to the initial alternatives—
BLOTTISHAM: Then we have genuine information loss.
QUILLIBRACE: And therefore a serious challenge to unitary quantum evolution.
BLOTTISHAM: So the word "lost" conceals an entire hierarchy.
STRAY: Yes.
BLOTTISHAM: And the paradox is really about which level of that hierarchy nature permits.
QUILLIBRACE: Exactly.
There was a silence.
Then Blottisham frowned.
BLOTTISHAM: But we've only moved the problem.
QUILLIBRACE: Naturally.
BLOTTISHAM: We now have information being reorganised into correlations in the radiation.
STRAY: Yes.
BLOTTISHAM: Fine.
But how does that happen?
QUILLIBRACE: An excellent question.
BLOTTISHAM: And where are those correlations encoded?
STRAY: Another excellent question.
BLOTTISHAM: And what has the horizon got to do with it?
Quillibrace folded his newspaper.
QUILLIBRACE: Now you are beginning to ask the right questions.
BLOTTISHAM: I thought you were going to say I was confused.
QUILLIBRACE: You are.
BLOTTISHAM: Oh.
QUILLIBRACE: But you are becoming confused in increasingly productive directions.
Elowen smiled.
STRAY: Which may be what progress looks like at the edge of a conceptual system.
Quillibrace glanced at the blackboard.
QUILLIBRACE: We have discovered that information need not be a thing.
That loss need not be destruction.
That invisibility need not be absence.
That scrambling need not be erasure.
And that preservation need not mean persistence of form.
BLOTTISHAM: Which leaves the horizon.
STRAY: Yes.
QUILLIBRACE: The boundary that is not a wall.
BLOTTISHAM: The place where nothing crosses, except apparently information.
QUILLIBRACE: And there you have already smuggled the next metaphor into the room.
BLOTTISHAM: Which one?
QUILLIBRACE: Crossing.
Blottisham looked at him.
Elowen looked at the blackboard.
Then she added one final word beneath the others:
BOUNDARY
STRAY: Perhaps our next question is not where the information goes.
She underlined the word.
STRAY: Perhaps it is what kind of thing a boundary is, when the thing being bounded is a set of possible causal relations.
Quillibrace picked up his newspaper again.
QUILLIBRACE: That should keep us occupied.
BLOTTISHAM: I had hoped for a simpler answer.
QUILLIBRACE: You have been given one.
BLOTTISHAM: What is it?
QUILLIBRACE: There isn't one.
BLOTTISHAM: That's not an answer.
QUILLIBRACE: It is an accurate description of the situation.
Miss Stray laughed.
Outside, the evening light had begun to fade.
The Common Room settled back into silence.
But on the blackboard, four words remained:
LOST.
HIDDEN.
TRANSFORMED.
BOUNDARY.
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