THE ORIGIN OF QUANTUM PROBABILITY
In the Quantum Entanglement page, we saw that Alice’s and Bob’s entanglement experiment combined two different states – one where Alice measures up while Bob measures down and the other where Alice measures down while Bob measures up. Once a measurement is made, only one of these two states collapses to reality, or so the Copenhagen interpretation goes. In fact, you don’t need entanglement to have a system in two (or more) states at the same time according to the Copenhagen school. However, it was Einstein’s publication of the EPR paper (see the same Quantum Entanglement page) that prompted Erwin Schrödinger to devise his famous thought experiment. This was a box containing a cat which was in the superposition of two states: an alive cat and a dead cat, depending upon whether a radioactive atom in the box had decayed and ultimately killed the cat. This ought to demonstrate what Schrödinger thought was the absurdity of such a notion.
So Einstein and Schrödinger were on the same side, against Bohr. They argued with Bohr that a cat couldn’t possibly be both alive and dead at the same time. Bohr, however, was unperturbed, replying that this was precisely what quantum mechanics said: until you collapse the wave function, the cat would be in the two very different states.
The Many Worlds interpretation (MWI) (see the Many Worlds page) nearly resolved the debate by saying that the two states, cat-alive and cat-dead, are both real and exist in different branches. However, as we saw in the Parallel Block Universes page, the topology of the Many Worlds branching tree isn’t consistent with special relativity’s requirement that we are in a non-branching block universe. So we have to upgrade our MWI model to a multiverse of stand-alone, parallel block universes, some of which contain the live cat and some, the dead cat.
This page describes this multiverse of parallel block universes in more detail.
In the final figure of the Parallel Block Universe page, I used eight parallel block universes to illustrate the entanglement experiment in the Quantum Entanglement page. In three of the universes, both Alice and Bob measure spin-up, in three of them, they both measure spin-down, in one, Alice measures spin-up when Bob measures spin-down and, in the last one, Alice measures spin-down when Bob measures spin-up.
In other words, the number of universes in which a particular event occurs is proportional to the probability of that event occurring, according to quantum mechanics. We are going to see how this works in practice.
Let us start with the simple Stern-Gerlach experiment shown in Figure 8a of the Quantum Spin Filter page. In this experiment, an electron whose spin has not been previously measured is passed through a spin filter. There is a 50% chance that the spin will be detected as “up” and a 50% chance that it will be detected as “down”.
Since the chances are equal, the two results can be captured as in Figure 1 by two parallel universes, in one of which the detected spin is “up” and in the other of which the detected spin is “down”.
Suppose that we now perform a second experiment which is identical to the first. If we perform it in the universe in which the result of the first experiment was “spin-up”, then there are two equal possible results for this second experiment: “spin-up” and “spin-down”.
In Figure 2 I have shown two parallel universes, one for each of these results. In both of these two parallel universes, I have shown the result of the first experiment as “spin-up”.
If we perform the second experiment in the universe where the result of the first experiment was “spin-down”, then, again, we need two universes for the two possible outcomes, and I have also shown these in Figure 2, this time with the result of the first experiment being labelled as “spin-down” in both universes.
In Figure 3 I have drawn the 16 parallel universes needed to show all of the possible outcomes of four experiments in succession.
Now that we have performed the experiment four times in each universe, we can see what we would calculate as the probability of an outcome of, say, “spin up” in any universe if we lived in that universe. If you inhabit the left-most universe in Figure 3, you will have found that each of your four experiments yielded a result of “spin-up”. With no other information, you conclude that the chances of a result of “spin-up” are 100%. However, if you look at the yellow table at the bottom of Figure 3, you will see that this finding – a total of four “up” spins in a given universe – occurs in only one of the 16 universes.
If we lived in the second universe from the left, we would find three “spin-up” results, and one “spin-down”. From the table, you will see that there are four universes where we would find three “spin-up” results: apart from the second universe from the left, the third, the fifth and the ninth also have three “up” spins. So, in these four universes, we would calculate that the probability of an up-spin result is three chances in four, or ¾.
As we see from the table, in six of the 16 universes, we find an equal number of up-spins and down-spins. We can plot these results in a histogram, which I have done in Figure 4.
I have copied the yellow table from Figure 3 into the top of Figure 4 and I have drawn arrows to connect the numbers in the table with the corresponding histogram blocks in the plot below. The position of each lilac histogram block along the x-axis of the plot is a measure of the number of up-spins in a given universe. Instead of labelling the x-axis with the number of up-spins, though, I have written the number of up-spins as a percentage of the total number of measurements. So, for example, in universes where we found three up-spins, I have written that as 75%, because three out of four possible results, ¾, were “up”.
Looking at the plot in Figure 4, we can see that, in six universes, the probability of finding “spin-up” is calculated as 50%, which happens to agree with the quantum-mechanical result. True, there are more universes in which the calculated probability does not agree with quantum mechanics, but, nevertheless, the calculation of 50% is a more popular choice than 0%, or 25%, or 75% or 100%, taken individually.
You may have realised that the plot in Figure 4 is just the binomial distribution. We can use the binomial distribution to calculate what happens when we do more experiments in each universe, and I have plotted these for sequences of 100, 1,000 and 10,000 experiments in Figure 5.
As the number of experiments grows, the width of the peak shrinks correspondingly. In other words, the greater the number of experiments, the greater the proportion of universes in which the inhabitants calculate a probability of around 50% for a “spin-up” result. (I’m saying “around 50%” because there will always be a proportion of universes where the number of “spin-up” results is close to, but not exactly, half of the total number of experiments. The estimation of the probability of “spin-up” gets more accurate as the total number of experiments gets larger.)
Incidentally, the total number of universes containing all possible combinations of results increases enormously as the number of experiments increases. So, for 100 experiments, the number of universes required to span all of the combinations is 2100. This number is about the same as 10 multiplied by itself 30 times, written 1030. The number of possible universes each containing 10,000 experiments is an unimaginably large 210,000, or 103,010. Because of the vast differences in the scales of the four plots shown in Figure 5, I have drawn them all to have the same peak height so that you can compare their sharpness more easily.
In order to talk about outcomes of experiments in parallel universes, it is useful to have a shorthand system. I suggest using the word “kernel” to indicate the smallest possible collection of parallel universes populated with outcomes according to the corresponding quantum-mechanical recipe. So, for example, where quantum mechanics predicts a 50:50 distribution of up-spin and down-spin results from the Stern-Gerlach experiment in Figure 1, we write the kernel as (u + d) (see Figure 6).
So we can recast, in the form of a rule, the comment I made above about the number of universes in which a particular event occurs being proportional to that event’s probability:
Parallel Universe Probability Rule
The probability of an event happening in a universe is the same as the proportion of times that a universe with that event appears in the kernel of possible universes.
Figure 7 shows the kernel for two Stern-Gerlach experiments performed one after the other in each parallel universe. Instead of using an ordinary multiplication sign, though, I have borrowed the “tensor product” symbol from mathematics (a circle enclosing a multiplication sign) to represent the process of multiplying the kernels. Using the tensor product symbol helps us to remember that this is not just multiplying ordinary numbers together, but that we are multiplying all of the components of a universe by all of the components of another universe.
It is easy to perform tensor multiplication with these simple kernels – they just multiply algebraically as you can see at the bottom of the yellow text block. Each of the three algebraic terms corresponds to results in one or more universes.
In Figure 8 I have returned to the sequence of four experiments featured in Figure 3. In the yellow text box, you can see that five terms are produced when the kernel is multiplied out. Each of these five terms corresponds to one or more of the 16 universes required to span all possible outcomes.
Up to now, I have only talked about an experiment where there are two possible outcomes, each equally probable. Now let us look again at the experiment in Figure 15 (a) of the Quantum Spin Filter page. In this experiment, remember, an electron with spin-up in the vertical direction encounters a spin-filter inclined at 60°. Quantum mechanics tells us that there is a ¾ chance of the electron emerging from this spin-filter with its spin in the spin-up direction, that is, with the spin inclined at the angle of 60°. There is a corresponding chance of ¼ that it will emerge with its spin in the spin-down direction.
The smallest possible collection of universes that spans these possibilities is four – three of them in which the electron emerges “spin-up” and one in which it emerges “spin-down”. I have shown these four possibilities in Figure 9. In this case, the kernel is slightly more complicated that the kernel for the basic, 50:50, Stern-Gerlach experiment, and I have written it in Figure 9 as (3u + d).
When this experiment is performed twice in sequence, 16 universes are required to span the possible outcomes. I have drawn these 16 universes in Figure 10, under the yellow text box showing the tensor product of the two individual kernels for the experiment. You can see that, as in the combined kernel, there are nine universes where the result is “up” in both experiments, six in which there is one “up” result and one “down” result, and one in which both results are “down”.
If we repeat this experiment many times, then we can make a plot like the one in Figure 5. Figure 11 shows what this looks like for the same numbers of experiments that we used for Figure 5. However, since the number of universes in the kernel is four rather than two as before, the number of universes for a sequence of experiments rises even more sharply with the number of experiments than it did before.
Once again, though, you can see that, if you perform the experiment often enough, then in most universes the inhabitants will calculate a probability for seeing an up-spin that is close to, or exactly equal to, ¾, or 75%. This, of course, is what you would expect from the kernel, in which ¾ of the total of four spins are “up”.
This makes the point, once again, that, as long as the proportion of “up-spins” in the kernel corresponds to the quantum-mechanical probability, then, when you measure the outcome of that experiment many times, most versions of you will find that same probability, or one very close to it.
Now look at the experiment in Figure 12.
At the top right, I have copied a schematic drawing of a methane molecule from Wikipedia. The bond angles in this molecule are just under 109.5° as shown. Quantum mechanics predicts that the probability of an electron passing through a spin-filter at this angle, having entered it with a spin upwards in the vertical direction, is one-third. (You shouldn’t attach any significance to the methane molecule – it just happens to have the angle I wanted for the result of one-third!)
The smallest number of parallel universes containing that proportion of up-spins is three, with a corresponding kernel of (u + 2d), as shown in the figure.
What happens when two different experiments are made in a universe? Look at Figure 13 where I have drawn a block universe where Ant makes a Stern-Gerlach experiment with a spin-filter angle of 60°. We already know from Figure 9 that the kernel for this experiment is (3u + d).
In Figure 14 I have drawn another experimenter, Bee, who is performing the “methane-angle” experiment outlined in Figure 12. His kernel, as we saw, is (Bu + 2Bd).
Figure 15 shows how these two experiments combine. The bottom row shows four parallel universes, in three of which Ant has measured “spin-up” and in one of which she has measured “spin-down”. This is as it should be, according to her kernel in Figure 13. In the first of these universes, at the bottom left, Bee has measured “spin-up”.
Notice that Bee doesn’t have to make his measurement at the same time as Ant (indicated by the fact that the event of Bee making his measurement is located higher than that of Ant – later in time – in the block universe). The order of events in the block universe depends upon the motion of the observer, as we saw in the page on Wave Function Collapse, where it was Alice who made the first measurement, or Bob, depending upon the observer’s relative motion.
In the next universe up, the left-most universe of the middle row, and in the top one, at the top-left corner, Bee has measured “spin-down”. So, in the first column of three universes, Bee has measured one “spin-up” and two “spin-downs”, just as he ought to, according to his kernel in Figure 14.
The same is true of the other three columns. So you see that the combined kernel for both independent measurements is a kernel of 12 universes as shown in Figure 15. Moreover, you can see how the kernels tensor-multiply – check that the four terms in the kernel, 3AuBu, 6AuBd, AdBu and 2AdBd, correspond to the 12 universes in the picture. This ensures that, if Ant or Bee performs their particular experiment many times over, then, in most universes, they will each find that their results are in line with the ratios in their kernels and quantum mechanics.
So this is how probability emerges in the multiverse of parallel block-universes. However, I need to add a bit about quantum entanglement. The experiments with Ant and Bee are independent: the outcomes of Ant’s experiments are not correlated with those of Bee.
There is another way to draw the combined kernel of Ant’s and Bee’s experiments. Look at Figure 16.
In this Figure, I have summarised the combined kernel in a 2 x 2 matrix. The column on the left shows Ant’s kernel multiplied by the “spin-up” component of Bee’s kernel (namely, “Bu”) and the column on the right shows her kernel multiplied by the “spin-down” component of Bee’s kernel (namely, “2Bd”).
Equally, the top row shows Bee’s kernel multiplied by the “spin-up” component of Ant’s kernel, and the bottom row shows his kernel multiplied by the “spin-down” component of Ant’s kernel.
In fact, the numbers in the squares are just the separate coefficients in the combined kernel.
Because of the way the numbers in the four matrix squares are calculated, you always find that the products of the numbers in diagonally opposite corners are equal (see Figure 17 for this example).
Now look again at the quantum entanglement experiment performed by Alice and Bob as shown in the Quantum Entanglement page and in the final figure of the Parallel Block Universes page. From the definition of a kernel as the smallest possible collection of parallel universes populated with outcomes corresponding to the appropriate quantum-mechanical recipe, the kernel for Alice’s and Bob’s experiment is (3AuBu + AuBd + AdBu + 3AdBd).
If we write these coefficients in a 2 x 2 matrix as before, we get the picture in Figure 18.
However, the products of the diagonal numbers are now 9 and 1: they are no longer equal, and, instead, we have the symmetry shown. This state of affairs is telling us that Alice’s and Bob’s measurements are not independent: they are, of course, entangled.
So, in order to find the combined kernel when the experiments include entanglement experiments, we have to tensor-multiply the tensor product of the “independent” kernels by the tensor products of the “entangled kernels”.
The experiments or events in Figures 13 and 14 were completely independent of each other. However, events may not be independent. To take a macroscopic example, in the universes in which Schrödinger’s cat is dead, there is a burial in the pet cemetery. In the others, there is not. Whether the event of a burial takes place, therefore, depends ultimately, in this case, upon the quantum event of a radioactive decay. Of course, many dependent events will not have such macroscopically remarkable differences, but the principle is the same – some events will only occur depending upon the outcome of another event.
From this page, then, you will have seen that Schrödinger’s celebrated cat is both alive and dead. In some parallel universes it is alive and in some it is dead. Some versions of you will open the box to find it alive; other versions of you will discover a dead cat. The likelihood of your finding a live cat is the ratio of the number of universes featuring a live cat divided by the total number of parallel universes featuring the cat alive or dead. This is the Parallel Universe Probability Rule that I wrote earlier. You can find that proportion from the kernel for that event which, in turn, is determined by Schrödinger’s equation for whether or not the poison bottle is opened by the radioactive trigger.
We have seen above that, simply to accommodate the different outcomes of a few experiments, the number of parallel universes has to be unimaginably large. It is the mathematical structure of the Plexus that is the origin of such enormity and complexity. Nevertheless, the number of universes in the multiverse is not infinite. We shall see why in the finite multiverse page, but first, we need to play with infinity.
Click here to go to [12] Playing with Infinity