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.