It is perhaps worth stressing the distinction between unpredictability and intractability. Unpredictability has nothing to do with the available computational resources. Classical systems are unpredictable (or would be, if they existed) because of their sensitivity to initial conditions. Quantum systems do not have that sensitivity, but are unpredictable because they behave differently in different universes, and so appear random in most universes. In neither case will any amount of computation lessen the unpredictability. Intractability, by contrast, is a computational-resource issue. It refers to a situation where we could readily make the prediction if only we could perform the required computation, but we cannot do so because the resources required are impractically large. In order to disentangle the problems of unpredictability from those of intractability in quantum mechanics, we have to consider quantum systems that are, in principle, predictable.
Quantum theory is often presented as making only probabilistic predictions. For example, in the perforated-barrier-and-screen type of interference experiment described in Chapter 2, the photon can be observed to arrive anywhere in the ‘bright’ part of the shadow pattern. But it is important to understand that for many other experiments quantum theory predicts a single, definite outcome. In other words, it predicts that all universes will end up with the same outcome, even if the universes differed at intermediate stages of the experiment, and it predicts what that outcome will be. In such cases we observe non-random interference phenomena. An interferometer can demonstrate such phenomena. This is an optical instrument that consists mainly of mirrors, both conventional mirrors (Figure 9.1) and semi-silvered mirrors (as used in conjuring tricks and police stations and shown in Figure 9.2.). If a photon strikes a semi-silvered mirror, then in half the universes it bounces off just as it would from a conventional mirror. But in the other half, it passes through as if nothing were there.
FIGURE 9.1 The action of a normal mirror is the same in all universes.
FIGURE 9.2 A semi-silvered mirror makes initially identical universes differentiate into two equal groups, differing only in the path taken by a single photon.
A single photon enters the interferometer at the top left, as shown in Figure 9.3. In all the universes in which the experiment is done, the photon and its counterparts are travelling towards the interferometer along the same path. These universes are therefore identical. But as soon as the photon strikes the semi-silvered mirror, the initially identical universes become differentiated. In half of them, the photon passes straight through and travels along the top side of the interferometer. In the remaining universes, it bounces off the mirror and travels down the left side of the interferometer. The versions of the photon in these two groups of universes then strike and bounce off the ordinary mirrors at the top right and bottom left respectively. Thus they end up arriving simultaneously at the semi-silvered mirror on the bottom right, and interfere with one another. Remember that we have allowed only one photon into the apparatus, and in each universe there is still only one photon in here. In all universes, that photon has now struck the bottom-right mirror. In half of them it has struck it from the left, and in the other half it has struck it from above. The versions of the photon in these two groups of universes interfere strongly. The net effect depends on the exact geometry of the situation, but Figure 9.3 shows the case where in all universes the photon ends up taking the rightward-pointing path through the mirror, and in no universe is it transmitted or reflected downwards. Thus all the universes are identical at the end of the experiment, just as they were at the beginning. They were differentiated, and interfered with one another, only for a minute fraction of a second in between.
FIGURE 9.3 A single photon passing through an interferometer. The positions of the mirrors (conventional mirrors shown black, semi-silvered mirrors grey) can be adjusted so that interference between two versions of the photon (in different universes) makes both versions take the same exit route from the lower semi-silvered mirror.
This remarkable non-random interference phenomenon is just as inescapable a piece of evidence for the existence of the multiverse as is the phenomenon of shadows. For the outcome that I have described is incompatible with either of the two possible paths that a particle in a single universe might have taken. If we project a photon rightwards along the lower arm of the interferometer, for instance, it may pass through the semi-silvered mirror like the photon in the interference experiment does. But it may not — sometimes it is deflected downwards. Likewise, a photon projected downwards along the right arm may be deflected rightwards, as in the interference experiment, or it may just travel straight down. Thus, whichever path you set a single photon on inside the apparatus, it will emerge randomly. Only when interference occurs between the two paths is the outcome predictable. It follows that what is present in the apparatus just before the end of the interference experiment cannot be a single photon on a single path: it cannot, for instance, be just a photon travelling on the lower arm. There must be something else present, preventing it from bouncing downwards. Nor can there be just a photon travelling on the right arm; again, something else must be there, preventing it from travelling straight down, as it sometimes would if it were there by itself. Just as with shadows, we can construct further experiments to show that the ‘something else’ has all the properties of a photon that travels along the other path and interferes with the photon we see, but with nothing else in our universe.
Since there are only two different kinds of universe in this experiment, the calculation of what will happen takes only about twice as long as it would if the particle obeyed classical laws — say, if we were computing the path of a billard ball. A factor of two will hardly make such computations intractable. However, we have already seen that multiplicity of a much larger degree is fairly easy to achieve. In the shadow experiments, a single photon passes through a barrier in which there are some small holes, and then falls on a screen. Suppose that there are a thousand holes in the barrier. There are places on the screen where the photon can fall (does fall, in some universes), and places where it cannot fall. To calculate whether a particular point on the screen can or cannot ever receive the photon, we must calculate the mutual interference effects of a thousand parallel-universe versions of the photon. Specifically, we have to calculate one thousand paths from the barrier to the given point on the screen, and then calculate the effects of those photons on each other so as to determine whether or not they are all prevented from reaching that point. Thus we must perform roughly a thousand times as much computation as we would if we were working out whether a classical particle would strike the specified point or not.
The complexity of this sort of computation shows us that there is a lot more happening in a quantum-mechanical environment than — literally — meets the eye. And I have argued, expressing Dr Johnson’s criterion for reality in terms of computational complexity, that this complexity is the core reason why it does not make sense to deny the existence of the rest of the multiverse. But far higher multiplicities are possible when there are two or more interacting particles involved in an interference phenomenon. Suppose that each of two interacting particles has (say) a thousand paths open to it. The pair can then be in a million different states at an intermediate stage of the experiment, so there can be up to a million universes that differ in what this pair of particles is doing. If three particles were interacting, the number of different universes could be a billion; for four, a trillion; and so on. Thus the number of different histories that we have to calculate if we want to predict what will happen in such cases increases exponentially with the number of interacting particles. That is why the task of computing how a typical quantum system will behave is well and truly intractable.