What Happens To Quantum Information Inside a Black Hole? (Script) (Patreon)

Meet Alice and Bob, famous explorers of the abstract landscape of theoretical physics. Heroes of the gerdankenexperiment—the thought experiment—whose life mission is to find contradictions in the deepest layers of our theories. Today our intrepid pair are jumping into a black hole. Again. Why? Well, to determine the fundamental structure of spacetime and its connection to quantum entanglement of course. 

In recent and less recent episodes we’ve seen how thinking about black holes leads to weird ideas. Like the fact that there's a minimum meaningful distance in space or that all of the information of a volume of space fits on its boundary. We also saw a simplistic manner in which an entire new spatial dimension can actually be encoded on that boundary, leading to the notion of the holographic universe. We even saw one idea for how gravity itself could emerge from the thermodynamics on that boundary. 

In a couple of recent episodes we talked about how the emergent dimension could be encoded in the scale of patterns on the boundary, but these days most physicists working in that area think there’s a better way to do that encoding—and that’s with quantum entanglement—the subtle correlations between the bits of quantum information that define the properties of particles. In fact, as we’ll see in upcoming episodes, it’s starting to look like the entire fabric of space may be emergent from entanglement—that space is sort of knit together from the network of mutual information of its contents.

This idea of the role of entanglement came from thinking about black holes in parallel to the stuff that led to the minimal Planck length and the holographic principle. In particular, in thinking about what happens to quantum information in a black hole. This is ultimately going to lead us to a deep connection between entanglement and wormholes in the ER=EPR conjecture. And that same line of reasoning is going to completely mess with our intuition about where quantum information exists, leading to black hole complementarity. 

We’ll get to these in upcoming episodes. Today we want to set up the contradiction that drives all of this. And for that we need Alice and Bob to jump into a black hole. Well, just Alice actually. She always was a bit more adventurous than Bob. So, Alice and Bob head to Sagittarius A*, the supermassive black hole at the center of the Milky Way. Alice suits up and prepares to jump. Bob prepares to watch from a safe distance. 

Alice carries a clock that emits pulses of high-energy gamma rays with a regular period so that Bob can track her position and also observe how her flow of time changes. Alice also carries a box in which a single electron has its spin frozen. This is our qubit—our bit of quantum information. In this case the information is about whether that electron spin is pointing up or down. The box also emits regular pulses of gamma rays so it can be tracked independently. Excited to be the first human to see below an event horizon, Alice high-fives Bob and leaps head-first towards Sag A*, and as she begins her descent she throws the qubit box to fall just ahead of her.

Let’s start with Bob’s perspective. He watches Alice and box fall, training his gamma-ray telescope on them to monitor their flow of time. Those gamma ray photons arrive with increasing separation due to gravitational time dilation, and at the same time the photons are gravitationally redshifted—stretched to longer wavelengths. Bob has also switched on detectors sensitive to all wavelengths from X-ray to visible to radio. He settles in, expecting to watch Alice slow down as she approaches the event horizon, to witness her seconds stretched to hours, years, millenia. That doesn’t happen. Instead, he observes Alice fall quickly, speed up, and then suddenly slow down and merge with the black hole’s event horizon.

So what happened to the time dilation approaching infinity at the event horizon? Well it’s true that for any black hole you can find a distance from the event horizon with any ridiculously high time dilation you want. Like, a millisecond stretching to 100 trillion years. But the thing is, a free-falling observer like Alice is moving very very fast by the time she reaches these points—she’s approaching the speed of light. The time dilation increases but the amount of time spent at each point decreases. Ultimately, the former wins and both Alice’s and the box’s descent seem to slow and freeze—but only really really close to the event horizon.

Bob sees the box slow and freeze first, which means Alice appears to catch up to the box. And Alice's head slows before her feet, so from Bob’s perspective she appears to flatten. Bob had been worried about the opposite effect—that Alice might get spaghettified—elongated by tidal forces pulling on her head harder than her feet. But that doesn’t happen here because the human-scale tidal forces for a black hole this massive are deep inside. Instead, by the time the top of Alice’s head is just above the event horizon, her entire body is just above the event horizon. This effect doesn’t have a term, but Bob wonders if pancakification might be appropriate. 

At this point, Alice’s photons are much fainter than the cosmic microwave background, and their wavelengths much longer. They’re barely detectable even by Bob’s giant radio antenna. And then he loses sight altogether. The last thing he sees is that all of Alice’s qubits along with the experimental qubit get scrambled over the entire event horizon. Now any light from Alice is indistinguishable from the black hole’s own Hawking radiation—and may be equivalent to it. Hawking radiation is the faint glow that every black hole produces due to its non-zero temperature. For Sag A* that temperature is 10^-14 Kelvin, resulting in Hawking photons with wavelengths of over 100 million kilometers.

Bob’s is desperate to not lose track of the experimental qubit. But the only way is to collect all future Hawking radiation emitted by the black hole. He begins the construction of a detector that will surround the entire black hole—a type of Dyson sphere. He also goes ahead and uploads his mind to the ship’s computer so that he can outlive this black hole, and he settles in for the long game.

In the meantime, let’s look at what happened to Alice from her own perspective. Plummeting towards Sag A* takes some hours, but that’s mostly far from the event horizon. She rapidly accelerates and the latter part of the journey wizzes by exponentially quickly. But there’s no g-force from that acceleration and no tidal forces from the black hole. She’s in freefall towards an event horizon vastly larger than she is, so she feels like she’s floating. This is a consequence of Einstein’s equivalence principle, which tells us that for an observer in freefall, their local patch of spacetime behaves exactly like space in the absence of a gravitational field. 

Alice sees the black hole beneath her as a vast black surface. Oddly, that surface appears to move away from her as she approaches. See, the “true” event horizon looks black from a great distance due to the infinite gravitational redshift of photons struggling to depart it. That redshift results from the full trajectory out of the black hole’s gravitational well. But Alice can still encounter photons that were emitted a hair above the event horizon because she’s deep in that gravitational well. Effectively, she’s coming to them rather than waiting for them to reach her. Also, as Alice approaches the event horizon, her own extreme velocity sort of de-redshifts the outward-pointing photons that she passes through. She sees the record of objects that fell ahead of her, including the qubit box. 

Now comes the critical part of the experiment from Alice’s perspective. At some point the qubit box crosses the true event horizon ahead of her, even if she sees no marker of that horizon in her reference frame. Does she still see the box? Yes. But the box she sees is the box as it was just before crossing the horizon. She can only catch photons emitted by the box from below the horizon if she crosses that horizon herself. 

But before she leaps down that rabbit hole, Alice, being curious, turns around to catch a glimpse of Bob’s spaceship. She’s hoping that her clock is so slow relative to the rest of the universe that she’ll see all of future history play out in fast forward. She is disappointed. She’s only able to perceive events corresponding to photons that can actually reach her. Her extreme time dilation relative to the outside universe is counteracted by the fact that she’s falling at close to the speed of light, so few future photons can catch up to her. So, yes she sees things speed up, and continues to monitor a fast-forwarded outside universe from deep inside the black hole. But photons from the far future, like when Bob builds his Dyson sphere, reach the event horizon long after Alice has merged with the black hole singularity. 

She’s a little sad to miss the rest of Bob’s long life, but Alice is first and foremost a scientist, so turns her mind back to the experiment and plummets through the event horizon. And nothing weird happens. Remember the equivalence principle. It tells us that there’s no experiment you can do or observation you can make on the local patch of space that distinguishes whether that patch is freefalling or free-floating. This principle is the bedrock of general relativity and physicists are very uncomfortable by anything that appears to violate it. Remember this because we may have to break this later, and it should make you uncomfortable also.

But assuming the equivalence principle holds, Alice can’t see things freeze at the event horizon. Not the qubit, not herself. Nor does she catch up to the qubit. Now beneath the horizon, the qubit is still below her and gaining ground towards the dark surface that also always appears below. But now that surface is growing and stretching around behind her also. The outside universe is a shrinking fisheye view above—Alice’s sense of “up” irises closed, and all directions become an enveloping “down”, leading to a terminus of space and time. She can’t ever leave to tell Bob the result of the experiment, but from her perspective it’s a huge success. She now knows with surety that black holes can swallow quantum bits.

Let’s go back to Bob. Remember he’s now in the business of catching Hawking radiation and looking for signs of the experimental qubit. Though if we’re being honest he’s also looking for signs of Alice. The black hole radiates. It would lose mass this way if it were for the fact that it eats more cosmic microwave background than it spews up. But after 100 billion years or so the universe has expanded enough to dim the CMB below the temperature of the black hole, and finally Sagittarius A* starts to shrink, assuming it doesn’t eat anything else.

Bob is aware that the black hole will eventually completely evaporate through this radiation. He’s also aware of the law of conservation of quantum information and the black hole information paradox from watching ancient PBS Space Time videos. He knows that quantum mechanics and perhaps his own heart won’t allow the erasure of Alice’s information nor the experimental qubit from the universe. There’s a risk of that happening if the black hole evaporates to nothing, and so, just like many long-dead 21st century physicists, Bob concludes that the information must be hidden in the Hawking radiation.

Time drags on and the Hawking radiation appears completely random. He’s had to build several star-sized computers—Matryoshka brains—to store and analyze this data. No big deal—he has nothing but time on his hands. After the black hole has lost half of its mass—10^87 years later, when incidentally all the stars have gone out and their stellar remnants have scattered to the void. It’s now just Bob and the black hole. Bob analyzes the next in the endless stream of information-barren Hawking photons, but this time he detects the first sign of a faint pattern. The subtlest correlation starts to emerge between all past radiation, hinting at some profoundly scrambled order in the randomness of the bits stuck on the black hole horizon. 

Newly excited by his Sisophean task, Bob continues to collect. Eventually he managed to recovers the object of the experiment—quantum bit, proving incontrovertibly that black holes do NOT swallow quantum information. And just before the black hole evaporates entirely, Bob also recovers all bits that were once Alice, revealing the true source of his long patience. He brings to bear the full might of the enormous black-hole-powered Matryoska brains and reverse-engineers all of Alice’s qubits to reconstructs her consciousness in digital form and the two live happily ever after, digital ghosts in a dead universe. How sweet.

This might seem like a happy ending, and it is for Bob and for the version of Alice encoded on the event horizon, but not necessarily for the version that fell in, nor for physics. Let’s get back to our experiment. What really happened to the qubit? Bob saw it freeze on the horizon, and then eventually catches it when it emerged as Hawking radiation. Alice saw it fall through the horizon and then merge with the singularity just before she did the same. 

In physics, we often make progress by identifying contradictions in a theory. This is perhaps the primary value of the gerdankenexperiment. So it seems we’ve identified a contradiction. Two different observers end up with irreconcilably different answers to the question “where is the qubit”. 

There’s a reason we used a qubit instead of just asking, for example, where is Alice? That’s because the qubit allows us to break very specific rules in quantum mechanics and so it sharpens our contradiction. I already talked about the prohibition against deleting quantum information, which is why Bob concluded that the qubit had to be preserved in Hawking radiation. But there’s another related rule—the no-cloning theorem states that quantum bits can’t be duplicated. If the bit escapes in Hawking radiation, how can it also be inside the black hole? 

The problem with deleting or duplicating quantum information is that it breaks what may be the most fundamental tenet of quantum mechanics—as foundational as the equivalence principle is to general relativity—and that’s unitarity. The quantum wavefunction describes probabilities. Probabilities have to add up to one. Add up all the probabilities of an object's possible position and they can’t add up to more than 100%—because what would that even mean? Deleting or duplicating a qubit breaks unitarity, and so is a no-no. It seems we have to either break unitarity by duplicating the qubit in the Hawking radiation, or by deleting it as the black hole evaporates. OR we have to break the equivalence principle by saying that Alice and her bit never get past the event horizon. 

To find out which is the case, you’re going to have to wait another week. Remember, Bob waited 10^87 years—a week isn’t so bad. Then we’ll start to see some of the possible solutions to this paradox, from black hole complementarity to idea of the firewall at the event horizon. And we’ll continue to sharpen the tools we need to knit entangled qubits into the very fabric of spacetime.