Wednesday, April 05, 2017

Causality and cloning by black holes

On his and his junior colleagues' blog, John Preskill describes a recent Stanford workshop where black hole information puzzle big shots met with quantum information folks:
Here’s one way to get out of a black hole!
If a black hole wants to hire you, I recommend you to treat the pledge above as oversimplified hype in order to avoid a bad career move. ;-)

We're promised that Maldacena, Stanford (coincidence, I hope: Leland Stanford only had one son, Leland Jr), and Yang will soon release a paper that should be related to the recent double-trace paper by Gao, Jafferis, and Wall. Various people were trying to make the wormholes in ER=EPR traversable etc. One can "almost" get there but something stops you from fully completing the task.

A big potential paradox that is being addressed by these thoughts is the cloning of information in quantum mechanics – which, under a certain scenario and with a certain interpretation, seems to materialize in the presence of black holes. Preskill talks about his and Patrick Hayden's paper that argued that after the Page time, the information can be extracted from the radiation quickly, and he seems most excited by the idea that a black hole entangled with a quantum computer is equivalent to a (short) wormhole.

Well, cloning is prohibited in quantum mechanics. In classical physics, it's possible to take some classical information – imagine a few bits in the RAM memory of a classical computer (although I am irritated how many people think that there's something special about the binary examples) – and copy it. Just think what the command "A=B" is doing in BASIC or whatever is your first computer language. ;-)

The command "A=B" simply takes the bits from the part of the memory reserved for the variable "B" and copies it to "A". Cloning like that cannot be done with full quantum, unmeasured qubits during a quantum calculation. Why? A qubit is described by the Hilbert space with basis vectors \(\ket 0\) and \(\ket 1\). Is there an operation that would map the initial state \(\ket\psi\) to \(\ket\psi\otimes \ket\psi\) describing two copies of the same bit?

There's obviously none because the transformation is bilinear, not linear, but all operations in quantum mechanics have to be described by linear operators. Or, if you wish, the mapping would have to be\[

\ket 0 \to \ket 0\otimes \ket 0\\
\ket 1 \to \ket 1\otimes \ket 1\\
(\ket 0 + \ket 1) \to (\ket 0 + \ket 1) \otimes (\ket 0 + \ket 1)

\] for it to be cloning. But the linearity implies, by adding the first two lines, that we should have\[

(\ket 0 + \ket 1) \to (\ket 0\otimes \ket 0)+ (\ket 1\otimes \ket 1)

\] without any mixed terms. So the cloning operation contradicts linearity. This simple result is known as the quantum xerox (or cloning) no-go theorem, or another permutation of these roots and words.

In the presence of an evaporating black hole, it seems that this paradox arises, however. Throw a qubit to a black hole – by a qubit, I literally mean a piece of the memory of a quantum computer. This qubit may be measured inside the black hole. But the black hole eventually evaporates and the qubit may also be measured by a complicated measurement+calculation of all the Hawking radiation that resulted from the dissolution of the black hole. So in some sense, we can measure the same qubit twice.

I personally think that one doesn't need any fancy ER-EPR correspondence and not even black hole complementarity to deal with this would-be paradox. The reason is that the observer who measures the information inside the black hole has no chance to get outside again. And escaping the black hole would be needed for him to measure the whole Hawking radiation. For this reason, it's totally possible for the observables inside and outside to have nonzero commutators,\[

[L_{\rm inside}, M_{\rm radiation}] \neq 0

\] without any interference between the two measurements. There is simply no observer who makes one of these measurements first to "collapse" his wave function and who needs to predict the second measurement. If you're inside the black hole, you can't ever get outside again. If you're outside and capable of seeing all the black hole radiation, the black hole is gone so you can't ever get inside it again. That's it.

Special complications may arise because for certain purposes, the Hawking radiation from the "half-entropy-evaporated" black hole is already sufficient for certain purposes. This moment – the Page time – when the initial black hole entropy dropped to 1/2 is important because the remaining black hole is (almost) maximally entangled with the radiation that is already out.

I feel that many people – arguably including John Preskill – are insufficiently careful about the question whose observer's wave function is being discussed. Some of them really sound as if they were thinking – like typical anti-quantum zealots – that the wave function is some globally valid set of classical degrees of freedom. Well, it's not. It's very important, especially in these black hole information puzzles, to appreciate that Bob (inside the black hole) may be Wigner's friend while Alice (outside the black hole) may really be Ms Alice Wigner (another sister of Eugene Wigner whom Wigner would donate to Paul Dirac as his wife, after Ms Margit Wigner). When it's so, Bob may see the wave function that he uses to collapse, but Alice doesn't, or vice versa.

The only thing that is needed for consistency of the quantum mechanical picture is the consistency of the description from the viewpoint of one actual observer. Different observers' descriptions don't have to be the same or "consistent" – that's what Bohr's complementarity principle is all about. The black hole complementarity principle is ultimately just a carefully applied Bohr's complementarity principle within the causal diagram of a black hole.

To what extent the ER bridges allow superluminal transfer

The ER-EPR correspondence tells you to believe that wormholes may exist but they must be non-traversable. You may get in but you can't get out. That's just like black holes. Einstein-Rosen bridge is such a non-traversable wormhole – unifying two black holes' interior into one. The ER-EPR correspondence tells you that the two observers who fall inside two very distant, perfectly entangled black holes may meet inside, before they are destroyed by the singularity.

We must understand the causal conditions of a wormhole. We usually paint the Penrose diagram – and I did so many times in the past, too. However, here we take a simpler picture. Look at the light cone above. The upper, future light cone looks like a popcorn cone, doesn't it? So exactly the volume where the popcorn sits, it's the black hole interior from which you can no longer escape again. The light cone geometry makes it clear why you can't get outside. You would need to move superluminally.

The origin of the coordinates in the light cone picture is just a special point inside the star, before it's completely clear that it has decided to collapse to a black hole. This point already belongs to the black hole interior even though you can't immediately see anything special about it. Only if you wait for the black hole to be born and get stabilized, it is possible to construct the precise location of the black hole horizon. Only with hindsight, it becomes clear that the point at the origin was special as the "seed" of Trump's Great Wall of Mexico, if you wish. Just to be sure, the singularity only develops much later: the whole popcorn (truncated) cone is in the past of the singularity.

Also, the black hole spacetime is heavily curved. So while the circumference of the horizontal ("now") circles around the cone gets arbitrarily high on the light cone picture above, the black hole horizon gets stabilized and no longer grows, having the fixed proper area \(A\) for a very long time. But that shouldn't prevent you from appreciating that the event horizon is still a null surface, just like the light cone in the Minkowski spacetime that is pictured above.

Fine. Imagine that you have two black holes that are separated by one million light years. So you should think of two very distant light cones like that. Assume that they correspond to two black holes that are perfectly entangled. This condition really requires that the two distant black holes must have been close (at least) 1 million years ago and the entanglement distance was just gradually stretched to these proportions by waiting.

The ER-EPR correspondence says that these two black holes share the interior. So if Alice and Bob – who are one million light years away from one another – jump into their black holes according to a pre-agreed algorithm, they may meet again for a while, before they die. (Yes, folks at Hollywood, I think it is a good idea for a movie. I guess that the two people love each other so much that both of them decide to make the irreversible decision to meet.)

Doesn't this freedom to meet violate some bans by special relativity? It doesn't. The funny thing is that the event horizon is a border crossing with some particular rules. Before you cross it – when you enter the future light cone on the picture from the outside – you have to sign this particular document:
I, Ms Alice Wigner, have freely decided to travel to the black hole or wormhole interior. I realize that my trip back could violate the U.S. laws and the laws of Nature and I therefore pledge to die inside and never get outside again.
Well, even if she doesn't promise the border patrol to die inside, she will die inside, anyway. Nature's law enforcement forces make sure that she does. While her contribution to the mass may get "recycled", I think that when it comes to her perspective as an observer, it's destroyed by the singularity and cannot be "continued" through the radiation – in this sense, the classical causal diagram should be trusted literally. But you may still be worried: When Alice and Bob talk to each other inside the wormhole, before they die, they may learn about some events that took place one million light years away from the place where they jumped to the black hole or wormhole (the distance is measured outside the wormhole – the wormhole is a shortcut that could allow you to say that the distance was smaller, too).

How does the spacetime around Alice's event horizon learn about the events that took place in Bob's galaxy during the recent one million years?

Well, the event horizon can't be just an innocent border crossing. It must be a particular one correlated with its distant cousin. Outside the event horizons, the degrees of freedom may look like field operators \(\Phi(x,y,z,t)\) that perfectly commute between Alice's and Bob's regions. The separate black holes may be assumed to be described by the \(N\)-dimensional Hilbert spaces of microstates. A priori, the two black holes may be independent and not entangled, so the Hilbert space is the tensor product \(\HH\otimes \HH\) of two copies of the black hole Hilbert space. Its dimension is \(N^2\).

The field operators \(\Phi(x,y,z,t)\) in Alice's region are basically \(N\times N\) matrices acting on her \(\HH\), and similarly for Bob. However, the black holes are maximally entangled, in the state\[

\ket{ER} = \frac{1}{\sqrt N} \sum_{i=1}^N \ket i \otimes \ket i.

\] The structure of this state is equivalent to the "unit operator" mapping Alice's copy of \(\HH\) to Bob's \(\HH\) or vice versa. However, operators should be linear in one ket-vector space and one bra-vector space. Here, we have two kets (or two bras, if you look from the opposite side). So it's not a regular unit operator. Instead, it is an operator mapping the ket space to the bra space: an antilinear operator \(J_{ER}:\HH_{A,\rm ket}\to \HH_{B,\rm bra}\).

I want to argue that the two event horizons are highly connected null surfaces that permute some extremely high-energy, short-distance degrees of freedom but are invisible to the low-energy degrees of freedom. When Alice (or Bob) crosses it, the permutation forces her (or him) to connect the Hermitian-like field operators \(L_{\rm Alice}\) that only act on the \(\HH_{\rm Alice}\) factor (or similarly for Bob) by the "diagonal operators\[

L_{ER} = L_{\rm Alice} \otimes {\bf 1} + {\bf 1} \otimes L^\dagger_{\rm Bob}

\] which is shared by both people. If you interpret \(L_{\rm Alice}\) to be a generator of \(U(N)\), the unitary group mixing the Hilbert space of her black hole's microstates, then the expression above is simply how the same generator acts on the representation\[

({\bf N}, \overline{\bf N}).

\] Assuming the entanglement, the two black holes' Hilbert space is simply interpreted as the "fundamental tensor times antifundamental" representation.

It's irresistable to argue that the operators like \(L_{ER}\) are more physically natural than the original ones once Alice (and Bob) are going to probe the degrees of freedom that do depend on the precise microstate – and the field operators inside the black holes do. Why? It's because these operators commute with the "antilinear unit operator" that maps the two Hilbert spaces. And because Alice and Bob – before they jumped to their black holes – knew that the two black holes were connected, they could basically figure out that the two black holes are invariant under an operator of the form \(J_{ER}\).

And the most important low-energy degrees of freedom are those observables that have vanishing or small commutators with the previously measured observables. In particular, for an operator, "low-energy" means having a "small commutator with the Hamiltonian". The entangled state of the two black holes is a singlet under \(U(N)\) – the diagonal group as explained above – so it is invariant under all the diagonal \(U(N)\) transformations and the measurements done inside the ER bridge should have this property.

It would be nice to have some more "unavoidable" explanation why Alice's and Bob's field operators "have to be" continued to the shared, "diagonal" ones and see how it works in some controllable examples of quantum black holes. I am imagining the black hole event horizon as something that may "permute all the points on it" – a domain wall moving by the speed of light that does so. Some "projection of a fuzzball" or something like that could be the "local" description of a generic microstate's horizon. There could be other descriptions, too.

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