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.