Black Holes: The Key to Understanding the Universe

12

The Sound of One Hand Clapping

‘Entanglement is iron to the classical world’s bronze age.’

Michael Nielsen and Isaac Chuang38

Physicists love a paradox. Perhaps unusually, they spend their professional lives searching for situations that will cause their world view to collapse, because deeper understanding may grow from the rubble. Good scientists don’t want their beliefs to be vindicated through research. They want research to generate new beliefs. The intellectual value of the black hole information paradox and the related affront to common sense delivered by black hole complementarity is that it forces physicists into a corner. For determinism to be preserved, there must be an error in Hawking’s calculation. If there is no error, determinism must be sacrificed. Along either path, new insights await. Hawking’s calculation, after all, rests on the very well understood foundations of quantum theory and general relativity.

What happens at the singularity of a black hole is beyond current understanding. According to general relativity, the singularity marks the end of time for anything unfortunate enough to meet it. It appears to be a place where matter ceases to exist. The inability to compute in the region of the singularity is a major unsolved problem in physics. Hawking radiation, on the other hand, is not a phenomenon that requires an understanding of physics close to the singularity. Hawking’s calculation never strays beyond what physicists refer to as ‘low-energy physics’ – the (apparently) well understood domain of quantum physics and relativity in the near-horizon region. As we shall see, it turns out that our familiar low-energy laws do contain traces of the deeper theory of quantum gravity and these traces are made visible by studying Hawking radiation.

The first key insight came from understanding the unusual way Hawking radiation is produced and the constraints this places on the evaporation process – in particular, the fact that Hawking radiation is ‘plucked’ out of the quantum vacuum. Take another look at Figure 10.1, which shows the emergence of a pair of particles from the quantum vacuum. Because these two particles have their origin in the vacuum, they share certain properties with it. Most importantly, they are quantum entangled.

Of all the bizarre aspects of quantum physics, none is perhaps quite so bizarre as the notion of entanglement. According to Schrödinger, entanglement is the phenomenon that forces the quantum world’s departure from classical thought, and it is the aspect of quantum physics that Albert Einstein referred to as ‘spooky action at a distance’. Entanglement isn’t considered so spooky today in the sense that it has become a part of our technology. It is the intellectual (and physical) resource that underpins the nascent quantum computers programmed and studied in many research laboratories. Entanglement is counter-intuitive, but it is also a tangible property of the world.

Entanglement has no counterpart in our everyday experience, which is why it appears counter-intuitive. Roughly speaking, it is a correlation between two or more things that is inexplicable using classical logic.* Entangled objects that are very far apart ‘feel’ each other’s influence instantaneously because they should really be viewed as a single connected system. This means that, underlying our everyday experience, there exists a more subtle, holistic world. There are electrons in your hand and electrons in the Andromeda Galaxy, separated by over 2 million light years, linked through quantum entanglement. This sounds like a near-mystical claim – spooky even. Crucially for the logical coherence of the world, however, these correlations cannot be exploited to send messages at faster-than-light speed, so don’t get too excited. Nobody will be using quantum entanglement to build time machines. Having said that, these remarkable correlations are real.

Qubits

To explore entanglement, we’ll introduce the notion of a quantum bit, or ‘qubit’. An ordinary bit is like a switch; it can only have two values, which we might call ‘on’ and ‘off’ or 0 and 1. These familiar classical bits are the basis of all modern computing. Qubits are a far richer resource because they can be both 0 and 1 at the same time. Whenever we measure the value of a qubit it will return either a 0 or a 1, but beforehand it can be a mixture of both. In the jargon, we say that the qubit is in a linear superposition of 0 and 1. If you’ve heard of the famous Schrödinger’s cat thought experiment, you’ll be familiar with this idea. A cat is sealed in a box, and it has been arranged (using a convoluted experimental setup involving decaying atoms and a vial of poison) that the cat is both alive and dead if the box remains sealed. When the box is opened, the cat will be observed to be either alive or dead. This is treating the cat like a qubit – it can be both 0 and 1 until observed. We won’t go into what constitutes an observation here, or why it may be appropriate to treat an object as large as a cat as a purely quantum system; for more detail you can read virtually any popular book on quantum mechanics, including our own, The Quantum Universe. All we need to know here is that qubits have a far richer structure than ordinary bits because they don’t have to be either 0 or 1; they can be both 0 and 1 at the same time.

Paul Dirac introduced a powerful notation to represent qubits and quantum states in general. Let’s consider a qubit which we’ll label ‘Q’. If it has a definite value of 1 then, in Dirac’s notation, we write:

|Q⟩ = |1⟩

If it has a definite value of 0 then we write:

|Q⟩ = |0⟩

An example of a qubit with an equal chance of it returning 0 or 1 when read-out (observed) is:

This has no counterpart in classical computing logic. A qubit that will return 0 for 10 per cent of the time, and 1 for 90 per cent of the time, is:

This is how the quantum rules work. We are to square the numbers to get the probabilities. This state is ‘mostly’ 1 with a small mix of 0. It’s worth emphasising that this qubit is not ‘secretly’ a 1 or a 0 and for some reason we don’t know which. It really is both 0 and 1 at the same time. This is very counter-intuitive, but it’s the way our Universe works.

Entanglement is a different but related idea. Imagine we have two qubits. If they are both 0, we could write their combined quantum state Q2 as:

|Q2⟩ = |0⟩|0⟩

where the first refers to the first qubit and the second refers to the second qubit. We could also imagine a state:

This is an entangled state.† There is a 50 per cent chance that the first qubit will have value 0 and the second qubit will have value 1, and a 50 per cent chance that the first qubit will have value 1 and the second qubit will have value 0. But notice that there is no chance that both qubits will be 0 or both qubits will be 1. A photon is an example of a physical system that behaves as a single qubit. It possesses a property known as spin, which can be either 0 or 1. The entangled ‘Bell state’ above can be realised as a system of two photons. States like this are routinely created in laboratories to study entanglement and for use in quantum cryptography and computing.

Let’s put these qubits aside for the moment, and switch to a wonderful analogy developed by Paul Kwiat and Lucien Hardy known as the quantum kitchen.39 Replace cakes for photons and change a few other words in what follows, and the story relates to real experiments that have been carried out in laboratories. The quantum kitchen is illustrated in Figure 12.1. The kitchen sits in the middle and two conveyor belts emerge from either side. Pairs of ovens travel along the conveyor belts and inside each oven is a cake that is baking as the oven moves. The cakes will be examined by Lucy (on the left) and Ricardo (on the right). The ovens can be opened halfway along, allowing the baking cakes to be observed. They will either have risen or not risen. At the end of the line, Ricardo and Lucy can make a different observation by eating the cakes. They will either taste good or bad. This is the experimental setup.

Lucy and Ricardo are going to encounter many pairs of cakes, and for each pair they will make a random choice as to whether to open the oven halfway along and look at the cake or to wait and taste the cake at the end. They are only allowed to make one observation each per pair of cakes; they can either open the oven halfway along and check whether their cake has risen, or they can taste their cake at the end, but not both. Their task is to record the results.

For the first pair of cakes, Lucy tastes her cake at the end of her conveyor and it tastes good. Ricardo does the same and finds that his cake tastes bad. For the second pair of cakes, Ricardo opens his oven halfway along and sees that the cake has risen. Lucy tastes her cake at the end and finds that it tastes good. And so on. After many cakes, they find:

Whenever Lucy’s cake rose early, Ricardo’s cake always tasted good.

Whenever Ricardo’s cake rose early, Lucy’s cake always tasted good.

In cases where both Lucy and Ricardo checked the ovens halfway along, 1/12 of the time both cakes had risen early.

If we use our common sense, honed by our experience of the world, these three observational facts lead us to infer that both cakes should taste good at least 1/12 of the time. We can infer this because:

when Ricardo and Lucy checked both ovens, they found that in 1/12 of the cases both cakes had risen, and

we know that when Ricardo’s cake has risen, Lucy’s tastes good and vice versa.

There is nothing surprising here, other than that they are rubbish bakers. However, here comes a shocking observation. In the quantum kitchen:

Both cakes never taste good.

How can that be? Using what seems like unimpeachable logic, we have concluded that in at least 1/12 of cases both cakes should taste good, and yet they never do.

It’s fun to play around and try to work out what is wrong with this reasoning – what possible mechanism could be in play to produce these strange results? Physics students like to do this in the pub. Long ago, the authors spent an evening figuring out why attaching a rod to the Moon and tapping it in morse code could not violate the principles of relativity by sending signals faster than light.

Figure 12.1. The quantum kitchen, from Paul Kwiat and Lucien Hardy’s ‘The Mystery of the Quantum Cakes’. (Figure 1 from ‘The Mystery of the Quantum Cakes’, by P. G. Kwiat and L. Hardy, American Journal of Physics, 68:33–36 (2000), https://doi.org/10.1119/1.19369. Reproduced here by permission of the authors and the American Association of Physics Teachers.)

There could be some mechanism whereby the only way to make a good-tasting cake is if the other person opens their oven at the midway point. Maybe the act of opening one oven midway causes a sound that shakes the other oven in just such a way as to make its cake taste good. Or perhaps a tiny, heat-resistant culinary gnome is sitting in each oven watching what is happening to the other oven who ensures that their cake tastes good if they see the other oven door opened. Both these possibilities (which exploit a causal link) can be ruled out if we make the conveyor belts sufficiently long and fast-moving that no signal can travel between the ovens before the observations are made. The signal would need to depart after Lucy/Ricardo has decided to open their oven at the midway point and arrive at the other oven before Ricardo/Lucy tastes their cake. We can arrange things so that there isn’t enough time for this to happen. Or perhaps the chef who makes the cake mixture somehow knows in advance the measurement choices that Ricardo and Lucy are going to make. The chef could then produce a bad mixture at just the right time. This possibility can be eliminated if Ricardo and Lucy make their random decisions after the ovens have left the kitchen. And so on. There is one logical possibility that could explain the results without resorting to quantum theory: every event in the Universe is predetermined, freewill does not exist and the results were baked in at the beginning of time. Putting that aside, we are left with quantum mechanics.

The results we’ve quoted above can be explained if the cakes are produced in an entangled quantum state. In this case, the quantum state of the cakes is such that both cakes can never taste good. This is like the situation we encountered earlier with our entangled system of qubits; there was no chance that both qubits could be 0 or 1.

Here is a quantum state for the cakes that reproduces the results obtained by Lucy and Ricardo:

where B and G mean ‘tastes bad’ and ‘tastes good’ respectively and the subscripts refer to Lucy and Ricardo’s cakes. This is a more complicated state than we’ve seen before, but you can see that, because there is no |GL⟩|GR⟩ term, both cakes never taste good. You can also see that both cakes taste bad one third of the time. To explain the numerical results associated with opening the ovens and observing whether the cakes have risen, we need a bit more quantum theory that won’t be necessary for what follows, but it is interesting, so we’ve moved the discussion to Box 12.1.

BOX 12.1. More from the quantum kitchen

You might be wondering where the ‘risen’ or ‘not risen’ measurements fit into the quantum state of the cakes. To reproduce the results obtained by Lucy and Ricardo, the ‘tastes bad’ and ‘tastes good’ states of the cakes are given by:

where R and N mean ‘risen’ and ‘not risen’. How should we interpret these states? Let’s take |B⟩ as an example. If a cake is observed to taste bad, then this means it must be in state |B⟩. If we now make a subsequent observation and ask whether it has risen or not, there is a 50 per cent chance it will not have risen, because 1/√2 squares to ½. If you fancy a little bit of mathematics, you can substitute these expressions for |B⟩ and |G⟩ into the state |Q⟩ to ascertain the coefficient of the |RL⟩|RR⟩ term. You should find it’s –1/√12 which gives a probability of 1/12 that both cakes will have risen. With less effort, you can also see that there is no |RL⟩|BR⟩ and |BL⟩|RR⟩ piece in |Q⟩, which explains facts 1 and 2. Our quantum chef is responsible for preparing the cakes in these very specific states.

For our purposes, the most important feature of the quantum kitchen is that, because the cakes are in an entangled state, they do not possess the qualities of ‘tastes good’, ‘tastes bad’, ‘risen’ or ‘nor risen’ independently. Rather the whole two-cake system is produced by the quantum kitchen in a state that mixes all these possible measurement outcomes with correlations that give the results we quoted above. And yet the quantum state is also such that in every case, each individual cake has the potential to taste good or bad, or to rise or not rise, when it leaves the kitchen. To reiterate a very important point, the probabilities observed by Lucy and Ricardo are not the result of a lack of knowledge about the state of the quantum cake system. They can know the state and yet before a measurement is made they cannot know whether an individual cake will taste good or bad, or is risen or not because each cake is all of these, until it’s observed.‡

Where does the information about the correlations in the entangled state reside? It is not stored locally by each individual cake. Returning to our simple two-qubit system:

a measurement of one of the qubits will reveal 0 or 1 with equal probability. That’s just like tossing a coin. Half the time it will come up heads and half the time it will come up tails. There is no information in that; it’s completely random. And yet, if the coins were quantum coins entangled in this way, if one of the coins came up heads, we would know for certain that the other came up tails. This would be true even if the coins were on opposite sides of the Universe. There is information stored in the state that prevents both the coins from ever coming up heads or tails at the same time, but it’s not stored in a way that is familiar to us. In a book, information is stored locally on each page and as we read more pages, the story gradually unfolds. In a quantum book each individual page would be gibberish and the story would reside in the correlations between the pages. We’d therefore have to get through a good portion of the book before we gained any insight into the story at all. No correlations, and therefore no information, would be discernible from a single page. The lack of information stored in a small part of a large, entangled quantum system is a very important property, which is central to the black hole information paradox.

Entanglement and evaporating black holes

The quantum vacuum is anything but empty. It is also heavily entangled. The extent to which the vacuum is entangled is captured by the remarkable Reeh–Schlieder theorem, which states that it is possible to operate on some small region of the vacuum in such a way that anything can be created anywhere in the Universe. This phenomenal conjuring trick is theoretically possible because the vacuum is inexorably entangled. The outlandish nature of the theorem is diminished ever-so-slightly by the fact that the local operation required is not something that we could ever perform, which is a shame. Nevertheless, the point remains that the vacuum has this encoding within it. Importantly for us, Hawking radiation is a product of this vacuum entanglement.

Figure 12.2. Entanglement between Hawking radiation and a black hole. At the top, how things proceed in the original Hawking calculation, where information is lost. Entanglement steadily increases between the radiation and the hole which causes a puzzle when the hole finally disappears (top right). At the bottom, how things will go if information does not get lost. The entanglement slowly transfers from being between the hole and the radiation to being entirely within the radiation.

Figure 12.2 illustrates the process of black hole evaporation by the emission of Hawking radiation. The dotted lines indicate pairs of Hawking particles that are entangled because of their origin in the quantum vacuum. Since the Hawking pairs necessarily straddle the event horizon, the entanglement can be pictured as being between the Hawking radiation exterior to the black hole and the black hole itself. As time passes, more and more Hawking radiation emerges, and more and more radiation particles become entangled with the black hole. But the black hole is shrinking. Eventually it disappears, and we have a problem, because the thing the Hawking radiation was entangled with has disappeared. The Hawking radiation left behind is like the sound of one hand clapping.

What are the consequences of the disappearance of entanglement? As we’ve seen, an entangled system has a rich structure that encodes information in the correlations across the system. That information must necessarily be lost if the entanglement is broken, which violates the rules of quantum mechanics and the basic principle of determinism.§ This is the essence of the black hole information paradox.

In order to dodge this unpalatable scenario, we might appeal to the fact that we do not understand what happens during the very final stages of black hole evaporation. When the hole is large, Hawking’s calculations are expected to be reliable because the spacetime in the vicinity of the horizon is not curved too much. This isn’t the case when the hole gets very small, just before it disappears. In these final moments, the spacetime curvature at the horizon becomes so great that we should not expect to be able to apply general relativity and quantum theory to make predictions. We are entering the realm of quantum gravity and, given that we don’t have such a theory, we might reasonably claim that all bets are off. Perhaps therefore, it would be wise to take a more cautious view and say that the information paradox may be resolved by some as-yet-undiscovered physics.

In 1993, North American physicist Don Page established40 that this line of reasoning is at fault and that appealing to unknown physics late in the black hole’s life doesn’t solve the problem because a paradox arises much earlier in the evaporation process, when the hole is middle-aged. Think of the black hole and the Hawking radiation together as a single entangled system that we are dividing into two; a more complicated version of the quantum kitchen. The black hole is one cake and the radiation is the other, and they are entangled. As more Hawking radiation is emitted, the hole shrinks with the result that more and more (radiation) is becoming entangled with less and less (the shrinking hole). There comes a point when the shrunken black hole no longer has the capacity to support the entanglement with the emitted radiation and, as Don Page realised, that will happen when the black hole is middle-aged.

We can illustrate Page’s reasoning with an analogy. Imagine a jigsaw puzzle made up of square pieces and imagine that the completed puzzle is set out on a table. The completed puzzle contains a large amount of information – the picture on the jigsaw. Imagine also a second empty table, onto which we will transfer pieces randomly selected from the jigsaw. The completed puzzle on the first table is like a black hole before it emits any Hawking radiation, and the empty table is the exterior of the black hole.

We now take a piece out of the completed puzzle at random and move it to the empty table, followed by another piece and then another. The pieces on the second table are like the Hawking particles emitted by the black hole. There are now three pieces on the previously empty table. These pieces are unlikely to reveal the jigsaw picture to us. If we focus only on these three pieces, we have no inkling that they are a part of a larger, more correlated and information-rich system.

The entropy of the pieces on the second table counts the number of ways we can arrange the pieces on the table if we pay no attention to whether the pieces fit together or not. This is like computing the Boltzmann entropy of a gas by counting arrangements of atoms (we are ignorant of the precise details). We will refer to this as the thermal entropy.¶

At first, when only a few pieces have been transferred, the entropy of table two increases with each piece we transfer because there are more pieces and we can put them anywhere we want. However, when a large enough number of pieces have been transferred to the second table, the pieces start to fit together. At some point, therefore, adding more pieces does not increase the number of ways we can arrange the pieces on the second table. Rather, adding more pieces leads to fewer possible arrangements as we start to see the bigger picture.

To make this more quantitative, we can introduce a new kind of entropy called the entanglement entropy. This entropy counts the number of possible arrangements of the pieces accounting for the fact that some pieces fit together. When only a few pieces have been transferred, the entanglement entropy is equal to the thermal entropy because none are likely to fit together. As more pieces are transferred, however, the entanglement entropy will eventually start to reduce because more and more pieces will fit together, restricting the number of possible arrangements. The thermal entropy, on the other hand, will continue to rise because it is concerned only with the number of pieces on the table.

The reason why this new quantity is called the ‘entanglement entropy’ is that it is a measure of how entangled the two jigsaws are. It is zero when no pieces have been transferred and it starts to increase as pieces get transferred. All the information contained in the jigsaw is still there, but it is now starting to become shared between the two tables. At some point, the entanglement entropy starts to fall again as the completed jigsaw begins to emerge on the second table. Information is now starting to appear on the second table and the amount of shared (entangled) information is falling. The two parts of the jigsaw are most entangled with each other when about half the pieces have been transferred, which is when the entanglement entropy is at its maximum value. For the jigsaw, therefore, the entanglement entropy starts from zero, rises to a maximum when both tables contain roughly the same number of pieces, and then falls back to zero again. We have sketched this in Figure 12.3.

Figure 12.3. The Page curve (black curve). Also shown is the Bekenstein–Hawking result for the entropy of a black hole (dotted line) and Hawking’s result for the entropy of the radiation (dashed line).

This simple jigsaw analogy provides a way to understand Page’s reasoning. The first table is analogous to the black hole and the second table is analogous to the emitted Hawking radiation. If information is conserved in black hole evaporation, we have just learnt that the entanglement entropy between the black hole and the radiation should first rise and then fall, as illustrated in the Page curve of Figure 12.3. The Page time is the time when the entanglement entropy stops rising and begins to fall. It marks the time when the correlations between the Hawking particles start to carry a significant part of the total information content of the original system.

We have also drawn the thermal entropy of the black hole, which gradually falls to zero as the black hole evaporates away, and an ever-rising thermal entropy for the Hawking radiation. This ever-rising curve is the result of Hawking’s original calculation, which appears to show that there are never any quantum correlations in the radiation. Page’s powerful point is that an information-conserving evaporation process must follow the Page curve and not Hawking’s ever-rising curve. And, crucially, the difference between the two curves manifests itself at the Page time, which is when the black hole is not too old and therefore quantum theory and general relativity should both be valid. Viewed this way, solving the information paradox is tantamount to understanding which curve is correct: the Page curve (information comes out) or Hawking’s original curve (information does not come out).

The Page time can also be thought of as the time at which we could begin to decode information contained in the Hawking radiation (if the information comes out). If the radiation is thermal, as it is to a good approximation before the Page time, then its entanglement entropy is equal to its thermal entropy. In this case, no correlations are visible and no information is contained in the radiation. After the Page time, correlations appear and the radiation becomes increasingly information rich. This is the situation we previously described for a quantum book. The initial pages are complete gibberish because the story is encoded in the correlations between the pages. It’s only when we get more than halfway through the book that we can begin to identify the correlations and decipher the meaning.

The fact that we have to wait until the Page time for any correlations to appear leads to another quite bamboozling idea. Because the Page time is roughly halfway through the lifetime of a black hole, which may be in excess of 10100 years, we are claiming that correlations appear between particles whose emission is separated by 10100 years. This illustrates the strangeness of quantum entanglement and perhaps hints that space and time may not be what they seem.

The conclusion we are forced into is that if information is to be conserved, some correction to Hawking’s original calculation must be present not much later than the Page time, which is approximately halfway through the black hole’s lifetime. But to reiterate a crucial point: at the Page time we expect both general relativity and quantum theory to be perfectly adequate in the near-horizon region; we would not expect to need presently unknown physics. And yet Hawking’s calculation, based on quantum theory and general relativity, diverges from expectations if we believe that information should be conserved in black hole evaporation. The challenge is now clear. If we are to show that information is not destroyed by black holes, we must calculate the Page curve.

We now find ourselves in the position of the theoretical physics community around the turn of the millennium. Don Page had laid down the gauntlet because the Page curve should be calculable using the known laws of physics. But the state-of-the-art calculation at the time was Hawking’s, which did not follow the Page curve. There was also complementarity, the crazy idea we met in the previous chapter, which lacked a convincing proof. Complementarity offers a solution to the information paradox in the sense that no information ever actually falls across the horizon and into the black hole as viewed from the outside – but we are also asked to believe that there is another point of view in which information does fall in, and that both points of view are equally valid. Around this time, a bold new idea surfaced that was crucial in convincing many that Hawking must be wrong, and that complementarity has substance. The key idea? The world is a hologram.

* Correlations are commonplace in everyday life: the colour of your left sock is likely to be correlated with the colour of your right sock and living in Manchester is correlated with experiencing drizzle.

† This entangled state is an example of what is known as a ‘Bell state’, named after Northern Irish physicist, John Bell, who pioneered early studies into quantum entanglement.

‡ It is an important feature of quantum mechanics that this ‘link’ between the cakes cannot be used to transmit information faster than light. We could ask for the probability that Lucy finds a good-tasting cake – something that Lucy can measure independent of Ricardo. We will discover that the odds Lucy observes do not depend upon Ricardo’s measurements, which means that Ricardo cannot use his choice of measurement to transmit a message to Lucy. Even though their observations are correlated, it is not a correlation that can be used to transmit information.

§ In quantum mechanics, we use the word determinism to refer to the fact that we can predict the future state of a system if we know its prior state. However, since quantum mechanics is inherently random, knowing the state does not also mean we know the results of experiments. In this regard, quantum mechanics is not deterministic. The more precise terminology is to say that quantum states undergo ‘unitary’ evolution.

¶ For example, if the jigsaw is a 3 x 3 puzzle with square pieces, where the pieces occupy one of 9 possible positions on a grid, the number of possible arrangements is 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 x 49 = 95,126,814,720. The thermal entropy of the jigsaw is the logarithm of this number. The factor of 49 is because each piece can be placed in one of four different orientations.