Probability in a causal universe

Does God play dice with the universe?

Probability is a familiar idea. When we roll a dice, we know there is a 1/6 chance of seeing a particular number between 1 and 6.

While we often say that the number we see came up by chance, we also know that that description really just reflects our lack of knowledge about the physical forces that acted on the dice. If we knew precisely its initial orientation, how fast it was rolled, what the surface characteristics were and so on, we could accurately predict which number would come up.

However the situation is more complex with quantum mechanics. An observer could have complete knowledge about a particular system, but not know which state they will measure it in. For example, suppose a photon is sent through a balanced beam splitter. The system evolves into a superposition of two relative states. In one state, the photon is on the reflection path, in the second state the photon is on the transmission path.

The Born rule states that the probability of measuring a system in a particular state is given by squaring the magnitude of the amplitude for the state. In the beam splitter example above, the amplitude for each relative state is √(1/2).^[1] So the probability for measuring each state is 1/2.

If the observer has complete knowledge about the system, it would appear that the probabilities are intrinsic to nature. It was this idea that Einstein disputed with when he famously proclaimed that God does not play dice with the universe. As he put it in a letter to Max Born, "Quantum theory yields much, but it hardly brings us close to the Old One's secrets. I, in any case, am convinced He does not play dice with the universe."

So in many textbooks the Born rule is stated as a fundamental postulate of quantum mechanics and not as something that needs to be explained. But it need not be this way. The Everett (Many Worlds) interpretation of quantum mechanics takes Einstein's side on the issue and requires that the Born rule be derivable from quantum mechanics, not merely postulated. If this succeeds, it returns us to the original understanding where probability is a result of a lack of knowledge. The twist, though, is that the observer could still have complete knowledge of the system under observation, but just not of their own location with respect to that system. I'm going to outline a derivation below that draws from Carroll's and Sebens' derivation.

1. On the Everett interpretation, measurement leads to initial self-locating uncertainty. An observer can have complete knowledge about the relative states of the system, but not which particular state they have just measured (since there is a version of them that has measured each state). This raises the question of how to quantify their uncertainty in terms of probabilities.

2. If the state amplitudes are equal, the observer should initially be indifferent about which state they have measured. So the states can simply be counted to calculate the probability that a particular state has been measured.

3. If the state amplitudes are not equal, they can be mathematically factored into states that do have equal amplitudes. And again the states can be counted to calculate the probability. The number of factored states exactly tracks the square of the initial amplitude, so it is equivalent to applying the Born rule.

This last point is interesting. Suppose that the wave function for a particular beam splitter gives a (non-normalized)^[2] amplitude of 1 for the reflection path state and an amplitude of 2 for the transmission path state. The Born rule says that the probability of observing the two states is in the ratio 1:4. That is, it is not correct to just count the states, or even just apportion the amplitude. Instead the amplitudes must be squared. But why should this be the rule?

This can be demonstrated by transforming the initial setup into a new setup with equal amplitude states as shown in Diagram 1. To do this, add a second beam splitter (with 1/2 probability of reflection and transmission) to the transmission path of the first beam splitter. Note: the amplitude for the reflection and transmission relative states is √(1/2) each.

Diagram 1: Factoring beam splitter states (amplitudes shown)

When a photon is sent through this setup, there are two new relative states on the transmission path with an amplitude of √2 each (i.e., 2 * √(1/2) = √2). Now add two more balanced beam splitters on each of those two paths. When a photon is sent through this setup, there are four new relative states on the initial transmission path with an amplitude of 1 each (i.e., √2 * √(1/2) = 1). Now the amplitudes for all five final relative states are equal to 1 and the ratio of the initial reflection path states to the initial transmission path states is 1:4 as required by the Born rule.

This can also be understood geometrically as an application of the Pythagorean Theorem as shown in Diagram 2.

Diagram 2: Factoring beam splitter states
(squared amplitudes shown)

The lengths of the sides of the green triangle at the far left represents the amplitudes for the first beam splitter states. The reflection path state has an amplitude of 1 and the transmission path state has an amplitude of 2. Since the relative states are orthogonal, the triangle is right-angled. The hypotenuse represents the superposition state which has an amplitude of √5. The red numbers represent the squares of the sides (which is the non-normalized probability of measuring the state).

The second green triangle represents the second beam splitter states with a hypotenuse of length 2. Since this beam splitter is balanced, the shorter sides have the same length. Therefore, by the Pythagorean Theorem, 2² = a² + a²; 4 = 2a²; 2 = a²; a = √2. Each of these shorter sides now becomes the hypotenuse for two further right-angled triangles with equal length short sides. Again, by the Pythagorean Theorem, (√2)² = a² + a²; 2 = 2a²; 1 = a²; a = 1. Thus the amplitudes are all equal to 1 with a reflection/transmission ratio of 1:4 as required.

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[1] If the photon entered the front of the beam splitter then, due to a phase change, the amplitude for the reflection path would be -√(1/2). For the purpose of this post, assume that the photon enters the rear of the beam splitters and so both the transmission and reflection amplitudes will be positive.

[2] Amplitudes are usually normalized such that the probabilities (the squares of the amplitudes) of the relative states sum to 1.