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.