The French scholar Pierre-Simon Laplace once told the story of a demon. The demon knows all the laws of physics, and is so smart that he can do an infinite number of calculations in his head. If you told him the exact state of the universe at one point in time, then he would be able to predict with certainty the exact state of the universe at some later time. He would always win bets.

He could also use his physics knowledge to turn the clocks back, and deduce, given the state of the universe at some time, the state it had at some earlier time. If you wanted to destroy a document containing information you’d rather no one ever find out, and, say, burned it, you still wouldn’t be safe. The demon could look at the smoke coming off the flames, and use it to deduce what was on the page.

Laplace told this story in order to convey the idea that

“We may regard the present state of the universe as the effect of its past and the cause of its future.”

This seems like a pretty sensible way to view nature to most physicists. The universe is in principle predictable. If it wasn’t the case, what’s the point in physics?

Fig 1: Given everything that’s happening at the present, one can in principle predict the future or deduce the past.

I’m going to tell you about a recent(ish) strange discovery that causes problems with this way of thinking. It concerns the bat-shit behaviour of black holes, and is referred to as the black hole information paradox.

Entanglement

We first have to understand a wee bit of quantum mechanics. The main thing about quantum mechanics is that physical things can exist in superposition. This is when the system exists as a mixture of different states that we usually would consider to be mutually exclusive, i.e., it only makes sense if it’s in one or the other.

For example, consider a single particle flying along through space. It can exist as a mixture of, say, an electron and a positron (the positively charged version of the negatively charged electron). It could be just as much positron as electron, or mostly electron and only a little bit positron, or the other way around. The quantum state of the particle can be encapsulated in one number ψ, telling you where it lies on the spectrum between electron and positron. ψ = 0 means it’s an electron, ψ = 1 means it’s a positron, ψ = 1/2 means it’s half and half.

What do I mean when I say it’s a mixture of electron and positron? Imagine the particle hits a detector, which can be used to deduce its charge. When it hits the detector, and the reading comes up on a screen, it needs to make up its mind. The chances of the detector registering a positron is ψ, and the chances of it registering an electron is 1-ψ.

Now let’s complicate the picture a little. Let’s say there are two such particles, call them A and B, which both sprang from the breaking up of some original particle. The original particle had zero electric charge, so the charge of A and B need to add up to zero. Both are in a superposition, both a mix of electron and positron. But, the requirement that their charges add up to zero limits the quantum states they are allowed to have. If particle A is an electron (negative charge), then B must be a positron (positive charge), and vice versa. They can’t both be electron or both be positron, as that would mean the overall charge not adding up to zero.

Both particles have a number specifying their quantum state; ψ A and ψ B . But this time, due to the requirement of overall zero charge, ψ A depends on ψ B , and vice versa. You need to know what ψ B is to know what ψ A is. A and B are said to be entangled.

If you left particle B out of the picture, then the quantum state of A is not well defined. It would seem like there is information missing from its quantum state, that information is being held hostage by particle B.

Let me elaborate on this a little to show what I mean by missing information. If we told Laplace’s demon the quantum states of A and B (i.e. the values ψ A and ψ B ), he could use the laws of quantum mechanics to predict exactly what their quantum states would be at some later time. However, what if we were only interested in particle A? What if we wanted to only tell the demon the quantum state of particle A, and ask him to deduce its quantum state at some later time? This couldn’t be done since particle A has information missing from its quantum state, so he couldn’t work out what would happen to particle A in the future. If the demon can’t see particle B, then his powers of perfect prediction are lost.

Fig 2: If you only know about particle A at time 1, this isn’t enough to predict its state at time 2. Only if you know the state of both particles at time 1 will you be able to predict either’s state at time 2.

This is kind of weird, but it doesn’t get in the way of Laplace’s belief. As long as the demon is given all the information available in the universe at a given time (which includes the states of both particles A and B) he can make perfect predictions of the future and deductions of the past. However, what if there was a way to, not just ignore the information in particle B, but physically destroy it?

Evaporating Black Holes

Ok, black hole 101. When a star dies, it collapses under its own gravity into a very dense and compact object. Some of the more massive ones will collapse into something that’s almost infinitely small and dense. Such a thing is called a singularity. Its gravitational pull will be so strong, it will prevent even light escaping from it. Get too close to it, and it becomes physically impossible to escape. You can imagine a sphere around the compact object signifying the point of no return, this is called the event horizon.

The weird nature of strong gravitational fields can make particles seem to be created out of nowhere. At the event horizon, particles appear in pairs. One flies outward, away from the black hole, and the other falls inwards toward the singularity. These pairs are entangled in a similar way that particles A and B were entangled. The quantum states of the particles radiating out of the hole are dependent on the state of those falling into the hole, who end up hiding behind the event horizon.

The black hole is always radiating these entangled particles, an effect referred to as Hawking radiation. If something is constantly throwing out energy, it will eventually run out of energy, and disappear completely. The black hole will evaporate leaving only the outgoing radiation as evidence of its existence. Information about the radiation’s quantum state, that was being held inside the black hole, has now been obliterated. Could it have somehow escaped before the black hole disappeared? No, it’s impossible for anything to cross the event horizon from the inside to the outside.

We are left with only a cloud of radiation that has a poorly defined quantum state. In fact, it is extremely poorly defined. Since it was so strongly entangled with the interior of the black hole, it contains almost no information. Compare this radiation to the radiation coming from a star (light, radio waves etc). If Laplace’s demon could collect up all the radiation from a star, it could deduce exactly the nature of all the reactions going on inside the star that led to the emission of the radiation. This is because, while the radiation seems random and messy, there is in fact subtle features hiding in it, delicate interactions between the constituent particles that can be used to deduce the nature of their origins. In this sense, the light from a star contains information.

Hawking radiation is not like this. It contains virtually no information, it doesn’t just look messy and disorganized, it is intrinsically messy and disorganized. The demon could collect up all the radiation left from the black hole, but he couldn’t deduce anything about the black hole from it.

The Information Paradox

Remember that document you really wanted destroyed, so no one could ever see, or even deduce, the information on it? Throwing it into a black hole seems a sure-fire way of doing that. Any information that falls into a black hole is permanently erased, since the end-state of a black hole is Hawking radiation containing no information.

The current laws of physics, or even any conceivable law of physics we could come up with in the future, are powerless to deduce what was going on before the black hole formation, even given the exact state of everything after the black hole evaporates.

Fig. 3: If you know everything at time 2, this will not be enough to deduce the information on the incriminating document at time 1, since all you have is Hawking radiation carrying insufficient information.

This also causes problems in the opposite direction in time. It seems likely at the moment that the fundamental laws of physics are symmetric in time, i.e., behave in the same way going both forward and backward in time. A video of the moon orbiting Earth would look just as sensible if played in reverse, since the equations governing gravity look the same if time is reversed.

If this is the case, then the laws of physics must allow the reverse of black hole evaporation to take place, i.e. fig.3 but flipped upside-down. Such a thing may never have happened in the history of the universe, and may never happen in the future, but the point is that such an event is allowed to happen in nature. This event would consist of radiation clumping together to produce a reverse-black-hole, and totally unpredictable things falling out of it. Our knowledge of the universe before the creation of the reverse-black-hole would not be sufficient to predict what would fall out of it. It could be anything, a sperm whale or a bowl of petunias for all we know, and no law of physics could ever tell us why they appeared.

Again, this type of thing may never happen, but the fact that our current laws of physics seem to allow this type of thing is deeply troubling to physicists. If information can be destroyed in a process like the above, who’s to say there isn’t a plethora of other possible processes in which information is destroyed?

Is it really true that the universe is fundamentally unpredictable? The debate has been ongoing since this problem was first uncovered in the 70s. A number of solutions to this problem have been proposed, for example, modifying the physical laws to let the information in the black hole somehow escape. So far none of the solutions have been conclusively shown to work, so the debate continues.

Some of the most notable attempts at a solution include: black hole complementarity, the existence of firewalls at the event horizon, an appeal to the principle of holography from string theory, and most recently, the theory of supertranslations.

We may be a long way from solving this problem, but I suspect when it is finally solved, it will come with some dramatic overturning of some of the most deep-rooted ideas in physics today.

More on Entanglement

More on Black Holes

More on Hawking Radiation

Oh yea one more thing, this whole discussion can be formulated in terms of entropy. I wrote an article about entropy and its connection to information, which will appear on your screen if you click here.

An entangled particle is not just missing information from its description, it is intrinsically missing information, which is being held by its entangled partner. The particle has an associated entropy due to this missing information, called entanglement entropy.

Hawking radiation is intrinsically missing information, in fact it is missing any microscopic information, so it is a “physical macrostate“. It’s not just a gas that you can give a thermodynamic description of, it is literally only thermodynamic! Hence, all you can deduce about the black hole from its radiation is its temperature, total energy, stuff like that, nothing else. Universe be crazy.