Black Hole Complementarity Question

[PeterDonis]

In a thread some time back on black hole firewalls, one of the papers linked to was one by Bousso in which he argues that (as the paper is titled--note that this is a revised version, the original was quite a bit dfiferent) "Complementarity is Not Enough". I'm not trying to start a general discussion on the various points of view on this topic; I am looking for the thoughts of any experts here on one particular point in the paper that I'm not sure about.

On p. 2 of the paper, Bousso presents the argument for why, if the equivalence principle is correct, quantum states outside the horizon must be (nearly) maximally entanged with quantum states inside the horizon:

Let us take the infalling observer, Alice, to be small compared to $r_S$ . She enters
the near-horizon zone (of order rS from the horizon) in free fall at the time $t = 0$. Near
the horizon, on time and distance scales much less than $r_S$ , Alice can approximate the
metric as that of Minkowski space. Assuming that no matter is falling in along with
her, Alice should perceive the vacuum of Minkowski space on these scales.

Minkowski space can be divided into a left and right Rindler wedge. The vacuum
state is maximally entangled between fields with support in the two wedges [8]. Locally,
the black hole horizon can be identified with a Rindler horizon, and B can be identified
with the right Rindler wedge. Therefore, Alice must find any modes that are localized
outside the horizon to better than $r_S$ to be maximally entangled with similarly localized
modes A inside the horizon.

I've bolded the part I am not sure about. I agree that the region outside the horizon, B, can be identified with the right Rindler wedge. But the argument given in the above quote also requires that the region inside the horizon, A, can be identified with the left Rindler wedge, and that doesn't seem right to me. It seems to me that A should be identified with the region behind the Rindler horizon, which is *not* the left Rindler wedge; it's the causal future of the origin (i.e., of the point at which the Rindler horizon and Rindler antihorizon cross).

Am I missing something, or is this a real flaw in the argument?

 

[atyy]

Maybe his terminology is different from yours?

In his figure 1 on p12, A and B seem to be like what you expect?

 

[PeterDonis]

Maybe his terminology is different from yours?

AFAIK the "Rindler wedge" terminology is standard and means what I said it means in my OP. I'm looking through the paper by Unruh that Bousso references at that point (reference 8 of his paper), but I haven't yet found the statement about vacuum state entanglement that Bousso refers to, so I can't yet confirm that Unruh was using the Rindler wedge terminology the way I understand it to be used.

In his figure 1 on p12, A and B seem to be like what you expect?

As far as I can tell, the blue shaded region in the figure is the past light cone of the event where Alice hits the singularity (I think the diagram overall is a Penrose diagram since the singularity is a horizontal line at the top). The region inside the horizon (which is, I assume, the 45-degree line going up and to the right that hits the corner where the singularity ends and future null infinity begins) would then not be the left Rindler wedge; it would be, as I said, the causal future of the "origin" (which actually won't be present in this spacetime since the figure is, I assume, supposed to describe an actual black hole formed by gravitational collapse, not an idealized "eternal" black hole). In fact, the region corresponding to what I think is the left Rindler wedge is not even present at all in the spacetime of a black hole formed by collapse.

 

[PeterDonis]

I'm looking through the paper by Unruh that Bousso references at that point (reference 8 of his paper), but I haven't yet found the statement about vacuum state entanglement that Bousso refers to

Actually I think I have found it now. On p. 879 of the Unruh paper, Fig. 1 appears, showing regions marked R+ and R-, which correspond to the right and left Rindler wedges, and the accompanying text makes clear that the (nearly) maximal entanglement is between field modes in these regions. But the interior of the BH corresponds to the region marked F in Unruh's diagram, not R-.

However, the Unruh paper has considerable further discussion of the black hole case which does appear to talk about entanglement between states in the exterior and interior of the black hole--and he also mentions the second exterior region (Region III of Kruskal spacetime, corresponding to R- in his diagram), so he's clearly not confusing the two. But his discussion seems to be entirely concerned with maximally extended Schwarzschild spacetime, i.e., with an "eternal" black hole, not with an actual black hole formed by collapse. So I'm not sure exactly what to make of it in terms of the question in my OP.

 

[atyy]

I'm not sure what the resolution is, but Maldacena and Susskind seems to have a similar setup on p24 and Fig 16 with A and B being almost maximally entangled, similar to Bousso's A and B. Then they also mention A', which might be analogous to Unruh's field in the left Rindler wedge, which they describe "In the thermofield state, B is maximally entangled with a mode on the left side of the Penrose diagram, at a point obtained by rotating the diagram by 180-degrees about the origin. That brings us to the operator A' at time -t". They say A' evolves into A. I'm not sure why the entanglement is preserved by time evolution.

 

[PeterDonis]

Maldacena and Susskind seems to have a similar setup on p24 and Fig 16 with A and B being almost maximally entangled, similar to Bousso's A and B. Then they also mention A', which might be analogous to Unruh's field in the left Rindler wedge, which they describe "In the thermofield state, B is maximally entangled with a mode on the left side of the Penrose diagram, at a point obtained by rotating the diagram by 180-degrees about the origin. That brings us to the operator A' at time -t". They say A' evolves into A.

Hm, this looks interesting, I'll take a more detailed look.

I'm not sure why the entanglement is preserved by time evolution.

As I understand it, this follows from the unitarity of time evolution. (Strictly speaking, it only follows if you're using a no-collapse interpretation, but that seems to be pretty much the standard interpretation among the physicists who are working in this field.)

 

[atyy]

There's seems to be relevant material in this paper by Czech et al. In figure 2 they suggest that states defined only on the left and right Rindler wedges, when entangled give rise to spacetimes that include future and past regions too. If I understand them correctly, they support this argument in section 4 by showing that if one has unentangled modes on left and right Rindler wedges, the stress-energy tensor diverges at the horizon. But for certain entangled states like the Minkowski vacuum, this divergence disappears.

In section 2.1 they describe how Rindler and asymptotically AdS spacetimes, which Maldacena and Susskind use, are related.

Naively, could Bousso's and Maldacena and Susskind's setup be something like taking Czech et al's Eq 5 and 6, and just assuming the equations hold outside both Rindler wedges?

 

[cenzi]

Non-Eucludian stuff, interesting to know.

 

[PeterDonis]

Naively, could Bousso's and Maldacena and Susskind's setup be something like taking Czech et al's Eq 5 and 6, and just assuming the equations hold outside both Rindler wedges?

I don't think so, because it looks like Czech et al's Eq 5 and 6 are meant to show that any separable state that can be factored into a portion with support on the right Rindler wedge, plus a portion completely independent of the right Rindler wedge, has a stress-energy tensor that diverges on the Rindler horizon. (I assume a similar argument could be made for the left Rindler wedge.) So the point of that part of Czech et al's paper appears to be to show that any state that is nonsingular on Minkowski spacetime must involve entanglements between the right and left Rindler wedges, i.e., *can't* involve expressions like Eq 5 and 6.

What I'm still not clear on is how this kind of setup translates to entanglements between the right exterior region (right Rindler wedge) and the future interior (instead of the left Rindler wedge). I can see how time evolution of a state with support in the Rindler wedges could eventually produce a state with support in the future interior; but what restricts the entanglement to just between the future interior and the right Rindler wedge, as opposed to involving both Rindler wedges? Put another way, what *dis*entangles the states in the left Rindler wedge from the other two regions?

Maldacena and Susskind's paper seems to take the position that it depends--i.e., it depends on what the hypothetical observer in the left Rindler wedge, Alice, decides to do. That's interesting, but not very satisfying (to me, at least).

 

[atyy]

I don't think so, because it looks like Czech et al's Eq 5 and 6 are meant to show that any separable state that can be factored into a portion with support on the right Rindler wedge, plus a portion completely independent of the right Rindler wedge, has a stress-energy tensor that diverges on the Rindler horizon. (I assume a similar argument could be made for the left Rindler wedge.) So the point of that part of Czech et al's paper appears to be to show that any state that is nonsingular on Minkowski spacetime must involve entanglements between the right and left Rindler wedges, i.e., *can't* involve expressions like Eq 5 and 6.

Yes, I didn't mean that so literally. I meant perhaps by taking Eq 5 and 6 to be meaningful outside the left and right wedge, one could get modes in the future region, analogous to Maldacena and Susskind's time evolution of A' to A. Then these could be the basis of a state in which a mode in the right wedge was entangled with a mode in the future region, something like in Eq 7 and 25.

What I'm still not clear on is how this kind of setup translates to entanglements between the right exterior region (right Rindler wedge) and the future interior (instead of the left Rindler wedge). I can see how time evolution of a state with support in the Rindler wedges could eventually produce a state with support in the future interior; but what restricts the entanglement to just between the future interior and the right Rindler wedge, as opposed to involving both Rindler wedges? Put another way, what *dis*entangles the states in the left Rindler wedge from the other two regions?

Bousso does say something about modes that are "localized".

Also, I think if A and B are localized modes, and if they are maximally entangled, then B cannot be entangled (at the same time) with another system that is distinct from A, provided quantum mechanics holds. I think Bousso calls this monogamy of entanglement, and in his sketch the AMPS dilemma is that B seems to be maximally entangled with A and H, which is forbidden by quantum mechanics. So if one accepts quantum mechanics, and if B and H are maximally entangled, then B and A cannot be maximally entangled. In the Rindler space picture, the lack of entanglement between B and A presumably indicates a lack of entanglement between modes in the left and right wedges, which would produce the divergent stress-energy tensor mentioned by Czech, an indication of a firewall.

 

[atyy]

In order to state the AMPS puzzle, is it necessary for B and A to be maximally entangled? If B and H are maximally entangled, then B cannot be entangled with A at all, let alone maximally. We know that some entanglement between B and A is necessary to prevent the firewall, so couldn't maximal entanglement of B and H (due to the black hole being old) be enough to set up the puzzle?

Preskill's sketch of the puzzle, for example, doesn't seem to require B and A to be maximally entangled if there is no firewall. http://quantumfrontiers.com/2012/12/03/is-alice-burning-the-black-hole-firewall-controversy/

 

[PeterDonis]

I meant perhaps by taking Eq 5 and 6 to be meaningful outside the left and right wedge, one could get modes in the future region, analogous to Maldacena and Susskind's time evolution of A' to A. Then these could be the basis of a state in which a mode in the right wedge was entangled with a mode in the future region, something like in Eq 7 and 25.

Yes, I agree that that looks possible; basically, we have a state A' in the left wedge and a state B in the right wedge that are entangled. Then A' evolves to A, which is a state in the future interior. By unitarity, if A' and B are entangled, then so are A and B.

But there are still a couple of issues here. One is that, in the usual picture of how A and B get entangled, there is some state, call it A*, in the *right* wedge (i.e., the same exterior region B is in) which is entangled with B (for example, say A* and B are a pair of spin-1/2 particles with net zero spin, so their spins must be opposite). Then A* falls inside the horizon while B stays outside; i.e., A* unitarily evolves to A, a state in the future interior. Then, since A* and B are entangled, so are A and B. That seems OK, but then why bother bringing in all this stuff about the left wedge? Why is it even necessary?

(One possible answer is the Czech paper's statement that a separable state with one factor purely in the right wedge must have an SET that diverges at the horizon. See further comments below.)

Another issue is the state A'', which is supposed to be the state A', but time evolved in the left wedge *instead* of time evolved into the future interior. But are A" and A *both* supposed to evolve from A'? If so, A must undergo some kind of interaction in the left wedge that splits it into two pieces, one which evolves to A'' and one which evolves to A. How does that interaction affect the entanglement with B? Or is only one of the two time evolutions (A' to A'', or A' to A) supposed to take place, but we don't know which unless we know what Alice, the experimenter in the left wedge, decides to do? (This seems to be implied by some of the things Bousso says that I think I quoted in a previous post.)

Also, I think if A and B are localized modes, and if they are maximally entangled, then B cannot be entangled (at the same time) with another system that is distinct from A, provided quantum mechanics holds. I think Bousso calls this monogamy of entanglement

All of the papers on this topic seem to agree that "monogamy of entanglement" is valid, yes (and most of them use that same term for it). But that also raises a question regarding A'' vs. A both time evolving from A' (see above). If A'' and A are distinct, then they can't both be maximally entangled with B--yet they both evolve from the same system, A', that *is* maximally entangled with B. Does one of them get "disentangled"? How do we know which one? (Or do both stay entangled, with the state growing more complicated as these "splitting" events occur? See below.)

in his sketch the AMPS dilemma is that B seems to be maximally entangled with A and H, which is forbidden by quantum mechanics.

Yes, but again, I'm not seeing how all this stuff about the left Rindler wedge comes into it; the simple scenario with a system A* falling into the hole from the right wedge, where B is, seems sufficient to raise the dilemma. Once again, A* starts out maximally entangled with B; then A* falls through the horizon. If "No Drama" holds, then A* evolves into A, a state in the future interior that is maximally entangled with B; but if "No Information Loss" holds, then A* has to also evolve into H, a state in the right wedge (the Hawking radiation) that is maximally entangled with B. So we have two distinct states, A and H, that both seem to be maximally entangled with B, purely based on things that happen in the right wedge and the future interior.

In the Rindler space picture, the lack of entanglement between B and A presumably indicates a lack of entanglement between modes in the left and right wedges

But this is the part I don't see. I'm not sure now that it indicates a problem with the argument that "No Drama" and "No Information Loss" appear to be inconsistent, because I don't see how entanglement between A and B requires any entanglement, or lack thereof, between B and any state in the left wedge (which might not really exist anyway). Entanglement between A and B can come about from entanglement between A* and B, as I described above, if A* falls through the horizon from the right wedge.

I do see that Bousso appears to be using the left wedge for more than just showing why there has to be entanglement between A and B; he is also using it to argue for the global topology of a black hole spacetime being more complicated than we think, as a way out of the dilemma. But I don't see any discussion in this part of his paper about how a state, A*, in the right wedge that is maximally entangled with B, and then falls through the horizon, would affect things. If A* is maximally entangled with B, then it would seem that B can't be maximally entangled with any state in the left wedge, by monogamy of entanglement.

Here, though, is where what is said in the Czech paper might come in. If we start out with a state that has only A* and B entangled, both in the right wedge, with everything else separable, then the SET of the state diverges at the horizon. So perhaps what is really going on in such a scenario is: we have a state A' in the left wedge and a state B' in the right wedge, which are maximally entangled, in order that the global state is nonsingular. B' then undergoes an interaction that splits it into A* and B; A* falls through the horizon from the right wedge to the future interior and evolves into--well, we can't call it A, let's call it A#-- while B stays in the right wedge. Similarly, A' undergoes an interaction that splits it into A'' and A; A'' stays in the left wedge while A falls into the future interior. In this scenario, *all* of A'', A, A#, and B are in a single entanglement relation, and the global state is still pure and nonsingular.

The problem I see with this is: where does H come from? If the two Rindler wedges are truly disconnected, then we still have a problem: there's no way for the information in A* to get duplicated into H as well as into A#, and that's true regardless of how many other states we pile into the entanglement relationship with B. Perhaps that's why Bousso resorted to postulating "wormhole" type connections between the black hole interior and the Hawking radiation; that would mean the state H is not actually a distinct state from the "A" states; it *is* one of the "A" states (either A or A#, depending on which version of the scenario pans out), brought through the "wormhole" into the right exterior.

In order to state the AMPS puzzle, is it necessary for B and A to be maximally entangled? If B and H are maximally entangled, then B cannot be entangled with A at all, let alone maximally. We know that some entanglement between B and A is necessary to prevent the firewall, so couldn't maximal entanglement of B and H (due to the black hole being old) be enough to set up the puzzle?

If H is a distinct state from any of the A's, then yes--that's basically what I was saying above, that there's no way for the information in A* (or any of the A's) to get duplicated into H without violating the no-cloning theorem.

 

[atyy]

I think you are right that the AMPS argument doesn't require entanglement between the left (L) and right (R) wedges. The AMPS argument just needs entanglement between a mode A in F and a mode B in R, as could happen if A and B are a Hawking radiation "virtual pair", with B being the outgoing mode.

I think Bousso says L, instead of F, because he identifies the mode in L and F as the same mode, ie. maybe he is thinking of something like http://arxiv.org/abs/0710.4345 (p39): "We note that even though u, v are defined in R only by eq. (2.19), by trivial extension $\psi$_λ,R(u) is also defined in P since u is finite on the past horizon V = 0. Similarly $\tilde{\psi}$_λ,R(v) is defined as well in F as well as R." So in u v coordinates, the mode in L and F are "the same mode", while in t and x coordinates, the mode in L evolves into the mode in F.