## Thursday, April 25, 2013

### An apologia for firewalls

Anniversaries: there are lots and lots of birthdays and deathdays of mathematicians who influenced physics today, including Felix Klein, Siméon Denis Poisson, Andrey Kolmogorov, and Felix Berezin.
A decade ago, I would enthusiastically read many or most papers authored or co-authored by Joe Polchinski who would be a fountain of crisp, creative, and perfectionist physics. I may have voted for him as the world's #1 most clearly thinking physicist.

Sadly for me, I unregistered from the club of regular readers of his papers after I looked at several important places in this new AMPSS paper (one "S" was added):
An apologia for firewalls
Their (AMPS) July 2012 idea that black holes have to have a "black hole firewall" at the event horizon (which instantly kills an infalling observer) has faced lots of criticism. Now they (AMPSS) try to argue that this criticism was invalid and the "alternative explanations" can't work.

Some of the criticisms – like the important point by Varela, Nomura, and partly Weinberg that the equivalence principle should only be applied on the Hilbert subspace built upon a particular "classical history" while the black hole is described by a Schrödinger-cat-like superpositions of macroscopically distinct states – were totally ignored. Even in isolation, these remarks are enough to show that the original paper by AMPS is wrong. It seems that AMPS suffers from many independent errors, each of which is enough to invalidate the argument.

After I tried to read several other paragraphs in AMPSS that attempt to "debunk" some other criticisms and they didn't make sense to me, either, I decided it was time to stop. I won't be able to get anything useful from the paper because the paper seems to contradict the basics of what I consider rational thinking. It's surely not at the level of Lee Smolin yet ;-) but it still seems rather hopelessly wrong.

Let me focus on the "first counter-argument" by AMPSS by which they try to disagree with the essential criticism many of us have raised, namely that $$\tilde B\subset E$$. This statement means that the quantum information inside a black hole should be viewed as a subset of the information describing a broader Hilbert space remembering the early Hawking radiation. These two packages shouldn't be counted as independent.

AMPSS honestly describe what this $$\tilde B\subset E$$ is supposed to mean; and they even say that this was apparently the very purpose of the black hole complementarity principle from the very beginning. However, they present the following monologue to suggest that the black hole complementarity in this sense is impossible.

We've known this counter-argument of theirs for half a year because the authors of AMPS were sending it to everyone who dared to point their mistake out. The AMPS' and AMPSS' counter-argument seems as wrong today as it did last summer. Why do they think that $$\tilde B\subset E$$ is impossible? We learn the following at the bottom of page 4:
1. Violation of quantum mechanics. As discussed in Ref. [1 = AMPS], this idea runs afoul of one of the basic consistency checks for complementarity: if a single observer can see both copies of a bit, then there is cloning, and quantum mechanics has broken down. In the present context, Alice can remain outside during the early radiation and extract from $$E$$ via a quantum computation a bit $$e_b$$ that will be strongly entangled with the later Hawking bit $$b$$. She then jumps into the black hole, capturing the entangled bits $$b$$ and $$\tilde b$$ as she goes, and so possesses in her laboratory three bits with no sensible quantum description. Their entanglements violate strong subadditivity [4 = Mathur's 2009 review of information paradox].
That's it. Needless to say, this monologue by AMPSS lacks any logic. There is absolutely nothing wrong about a qubit – more generally, an observable given by an operator – that manifests itself in two or several ways. What we're saying by the proposition $$\tilde B\subset E$$ is that it is the same qubit so there's manifestly no cloning; in her "lab", she will have just two, and not three, independent qubits. A cloning means that there were two different qubits (i.e. a four-dimensional Hilbert space) that a hypothetical (impossible) machine is supposed to bring to the same state. Cloning is impossible but the black hole complementarity is exactly the statement that the cloning isn't there because the "qubits" are redundant – they are just different labels for the same qubit (the Hilbert space is two-dimensional).

The would-be paradoxical paragraph above suggests that Alice could make a measurement of the early radiation and calculate a qubit describing the black hole interior – and she would later perceive this qubit herself. What a horror! ;-) This "scary scenario" that AMPSS present as an inconsistency is called "prediction in physics". All predictions in physics have exactly the same form!

Let me tell you an example. Take a harmonic oscillator with the Hamiltonian$H = \frac{p^2}{2m} + \frac{m\omega^2 x^2}{2}.$ The Heisenberg equations of motion imply that an operator at time $$t$$ – imagine $$L=x$$ but it can be any other operator – depends on time in the following way:$L(t) = \exp(iHt) L(0) \exp(-iHt).$ Because the spectrum of the harmonic oscillator is equally spaced with $$\Delta E = \hbar\omega$$, we can see that for $$t=2\pi k/\omega$$ with $$k\in\ZZ$$, the first exponential is a $$c$$-number (phase) that commutes with $$L(0)$$ and cancels with the opposite phase so we have$L(t+2\pi k / \omega) = L(t)$ All the operators are periodic with the period $$2\pi/\omega$$. Well, the periodic motion of a harmonic oscillator is something we know even in classical physics – and many schoolkids know about this periodicity, too.

You may watch the animated GIF for a minute, an hour, but it will still be periodic.

We have infinitely many "copies" of the operator $$L(0)$$ which may be a "qubit". What a horror! Needless to say, there is absolutely nothing wrong about this situation. An observer may observe $$L(0)$$ and if she is familiar with the maths of the quantum harmonic oscillator, she may calculate that the operator $$L(2\pi/\omega)$$ is literally the same thing as $$L(0)$$. So if she measures $$L(2\pi / \omega)$$ i.e. $$L$$ at some later time again, she will get the same result as she did at $$t=0$$. The operators are literally equal. There's no way to avoid the fact $$L(2\pi/ \omega) = L(0)$$; it's a consequence of the Heisenberg equations of motion – damn laws of physics!

You may call them "copies" but they have the same matrix entries relatively to a basis of the Hilbert space. They are entry-by-entry the very same thing. So it makes really no sense to call them "copies". You may talk about them in several sentences, copy the sentences at many places, imagine various "visualizations" and "consequences" of these observables, and use several symbols for the operator, but mathematically speaking, they are one object. One operator. That's why the measurement is guaranteed to produce the same result. It's a demonstrable law of physics.

The case of the qubit $$b$$ in the would-be paradoxical situation described by the modest paragraph in AMPSS is completely analogous. The Heisenberg equations of motion just imply a map between the operators and qubits that the infalling observer may measure earlier (before she crosses the horizon, for example); and those that she can measure later. The later operators are – in general non-linear – functionals of the earlier operators. That's what the evolution in physics – Heisenberg equations of motion – means. (The transformation/evolution/encoding of the qubits in the presence of a black hole event horizon is far more complicated and hard-to-follow than it is in the case of the harmonic oscillator.) The infalling observer may or may not have a fast enough quantum computer to calculate the prediction. But if she has one, she will just predict what she will observe inside the black hole and the observations inside the black hole will confirm the prediction even if she tries hard to get a different outcome.

It's that simple.

Let me mention that if she measures some other operator $$K$$ after she calculates the prediction but before she measures $$L(2\pi/\omega)$$ (or, analogously the qubit inside the black hole) that is expected to confirm the prediction, she may affect and disrupt the later measured value of $$L(2\pi / \omega)$$. That could invalidate the prediction. But there's no contradiction because the measurement of $$K$$ changes the problem – it adds some extra complication (linked to the measurement of $$K$$) or a perturbation to the Hamiltonian (think about the harmonic oscillator) – so the correct calculation that had established $$L(2\pi / \omega)=L(0)$$ has to be modified and perturbed for it to remain correct. At any rate, correct predictions will be verified; incorrect predictions may fail but if predictions are incorrect, we can't say that we have isolated the right "copy" of the qubit (in normal terminology: we can't say we have found the right transformation translating operators at one moment to those at a later moment). There is no paradox in either case.

I would understand if this rudimentary error were made for an hour, perhaps a day, and then the erring people would just agree that there is no paradox here. But this has been going for almost one year and the amount of time wasted with this non-problem and unconstructive debates about it could easily reach decades or centuries, too. So I am not going to read Joe Polchinski's (and other AMPSS authors') papers related to the black hole information puzzle anymore. It would be just a way to get upset.

1. I am looking at the wikipedia discussion on no-cloning theorem:

"The state of one system can be entangled with the state of another system. For instance, one can use the Controlled NOT gate and the Walsh-Hadamard gate to entangle two qubits. This is not cloning. No well-defined state can be attributed to a subsystem of an entangled state. Cloning is a process whose result is a separable state with identical factors."

The separable states are by definition unentangled. If you had two copies of the same separable state, then any use of the state as a basis would require an additional index in any expansion. This is non-sensical. The transitive postulate of equality clearly tells us that if A=B, and B=C, then A=C. The firewall people appear to be making a statement that this is not true. Now it is possible that euclid is wrong, and this has been shown previously, e.g. non-euclidean geometry, but the argument being made has clear mathematical consequences that are being brushed under the table.

2. I have watched this video:
http://pirsa.org/13010021
As I have not followed the endless debate on firewalls, I have no idea what to make out of the claim made:

"The physical quantum gravity states described by these observables must be solutions of the spatial diffeomorphism and Wheeler-deWitt constraints, which implies that the state space does not factorize into a tensor product of localized degrees of freedom.
The "firewall" argument that unitarity of black hole evaporation is incompatible with a regular near horizon region is based on such a factorization, hence is not applicable in quantum gravity."

3. Do you think in 50 years, students taking string theory 101 will study the firewall issue the way we studied the twin paradox in SR classes?

4. Is the issue that there is no unitary evolution between $e_b$ and $b$? In the case given above there is an obvious unitary process that allows the state to be "duplicated."

The hard part is showing how such unitary evolution can be responsible for the AMPS scenario. It seems like it has to be true, but producing the actual state evolution so far seems to have evaded all authors.

5. Some progressive thoughts continuing with regards to the development of thought experiments and theoretical position......I do not think you should be to disappointed with Polchinski.

Polchinski's early contribution to microscopic blackhole helped to advance understanding...while the need to advance that thought experiment necessary too.

Best,

6. To be sure, the majority of research into string theory
is not focused on how the theory connects to the real world; rather,
most physicists are exploring questions at a more theoretical level.
Such formal work is necessary, because as noted above, we need a deeper
understanding to fully formulate the theory. Even the many theorists who
are interested in how string theory connects to the real world don’t
typically think much about what it means to test the theory.
Fortunately, an increasingly active group of “string phenomenologists”
are focusing on formulating a string-based description of the world and
testing that understanding. They are already making testable
predictions, and will increasingly do so.String Theory and the Real World by Gordon Kane

7. Lubus, in your infinite wisdom, can you kindly tell me if you think Palm theory has any relation to physics at all? I'm not saying it does, I'm asking, I do know that Hawkes processes model self-exciting point-processes and that there exists extensions to the quantum domain in the form of feedback and the "measurement problem" (I can find the paper if you want) and a Hawkes process is related to a Palm process... anyway I thought the idea of mass-stationarity and transport was a neat idea... no idea if any of it is 'physical' though.

http://digbib.ubka.uni-karlsruhe.de/volltexte/documents/1641947

8. In that 2010 piece Kane points to the suspected - just week ago positron excess - as additional support of SUSY. The main problem for ST I think is political HEP at 30 to 100 tev costs lots. Funding crackpots that say that ST cannot be experimentally tested is many times less expensive than HEP experiments. Politicians either believe the crackpots or pretend to.
I don't think one can correctly I'd this problem with left or right. I wish it were otherwise but it is so often easy to profit and lie with bad science because few of our citizens have accepted education

9. There is something I don't understand.

if we consider that a late-time radiation b is maximally entangled with Eb ⊂ E = early-time radiation, I think it is relatively to an observer at infinite.

If we consider that b' (outside black hole modes) is maximally entangled with a (inside black hole modes), it is relatively to a free infalling observer.

So I think it is a nonsense to say that b = b', because the observers are totally different.

In fact, I will see more : b' = Eb and a = b,

Eb, for the observer at infinity, appears as a (subset of the) early-time radiation, but for the free infalling observer, it appears as outside black hole modes b,

So b' = T(Eb), where T is a transformation from the observer at infinity to the free falling observer

b, for the observer at infinity, appears as a late-time radiation, but for the free infalling observer, it appears as inside black hole modes a.

So, a = T(b)

So the entanglement would be exactly the same, because the entanglement between b and Eb is the same that the entanglement between a = T(b) et b' = T(Eb).

It is just the same reality, seen differently by the two different observers.

10. I completely agree with you. The entanglement of the early and late Hawking radiation only makes sense from frames such as the frame of the observer at infinity; but the same observer just doesn't see inside the back hole so he can't discuss further hypothetical entanglements with modes inside the black hole.

This is really a way to phrase the main observation behind the black hole complementarity principle. There can't be any paradox - conflict with monogamy of entanglement etc. here - exactly because no observer should be using a Hilbert space that contains the late radiation and the modes inside the black hole at the same moment.

11. This is not the end of the argument, though. The whole problem is in producing the form of this transformation!

12. No, Clayton, it's not.

The form of this transformation is almost certainly extremely complex, undecodable in practice, and it depends on the description of quantum gravity / string theory we prefer.

More importantly, general questions about the redundancy of the information or complementarity are more important and they don't require one to find the explicit form of such a transformation. So almost no papers try to discuss such an explicit form.

Hawking radiation is extremely scrambled information about the initial state. In fact, black holes are among the fastest scramblers that are possible in Nature.

13. I have had a little more time to look this over now. perhaps the sign at

http://www.flickr.com/photos/khaz/562608618/

Should be given AMPS for their use.

14. I know I'm late but I wanted to ask about the univeristy of York's work on the firewall paradox: