# Fungibility in quantum mechanics

In The Beginning of Infinity David Deutsch explains some of quantum mechanics in terms of fungibility. Two or more things are fungible if there is no difference between them apart from the fact that there is more than one of them. In this post I explain some stuff about fungibility in quantum mechanics. I think the only knowledge you need is the ability to add and multiply, knowledge of what a square root is, and the knowledge that $\pi = 3.14159\dots$. Other stuff is explained in the post.

For example, for the purposes of settling debts, one dollar is fungible with another. If you pay a debt, the person you have paid can’t legally require that you pay him with a particular dollar. He can’t require that you turn out your pockets and say “I want this dollar and not that one.” You can also have a situation of diversity within fungibility. If you have a debt, then your creditor may own some of the dollars in your wallet and not others. There is no fact of the matter about which dollar he owns, but he owns some of them. So despite being fungible the dollars have diverse properties.

According to quantum mechanics, the whole of physical reality is a complex structure called the multiverse that contains, among other things, multiple instances of all the objects you see around you. (Many physicists say that quantum mechanics isn’t about the multiverse, but they’re wrong. For an explanation of why they are wrong see Chapter 2 of The Fabric of Reality by David Deutsch and Chapters 11 and 12 of The Beginning of Infinity.) Some of those instances are different from one another. For example, some instances of me phrased the sentence before this one in a slightly different way because they had different ideas about how to write this post. But some of the instances of an object are physically identical except for the fact that there is more than one of them: they are fungible. In addition, you can have diversity within that fungibility (from BoI, p. 289, location 5011 in the Kindle edition):

Furthermore, it follows from the laws of quantum physics that, for any fungible collection of instances of a physical object, some of their instances must be diverse. This is known as the Heisenberg uncertainty principle’, after the physicist Werner Heisenberg, who deduced the earliest version of quantum theory.

Dollars are fungible as a matter of convention, not as a matter of physics. So with the dollars it is relatively easy to understand the fungibility. But how can it be the case that objects are physically fungible and how does quantum mechanics describe that fungibility?

First, what does it does it mean for two or more instances of a physical system to be fungible? It has to mean that everything about those systems that could be measured is identical. Each system exists in multiple versions that can interfere in interference experiments. Suppose there are two possible outcomes of a measurement of a particular physical quantity A of some system, and that the possible outcomes are 0 and 1, and that all the instances of the system start out with the value 0. The collection of the instances with the value 0 is represented by the state $|0\rangle$. There is also a state that would describe a collection of instances all of which have the value 1: $|1\rangle$. States can be added together by the following rules. (1) If you add two states that are the same their magnitudes can be added up so
$\tfrac{1}{2}(|0\rangle+|0\rangle) = |0\rangle$.
If the states are different then their magnitudes can’t be added up so $\tfrac{1}{2}(|0\rangle+|1\rangle)$ can’t be written in any simpler way.

There could be another physical quantity B with possible measurement outcomes $+$ and $-$, where
$|+\rangle =\tfrac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$

and
$|-\rangle =\tfrac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$.

Using the rules I described for adding up states you can work out that
$|0\rangle = \tfrac{1}{\sqrt{2}}(|+\rangle+|-\rangle)$.
Before I can describe the next thing I have to address a possible misconception. Quantum mechanics is commonly described as a probabilistic theory. What people usually mean when they say this is that if you measure a physical quantity one of the outcomes happens randomly with some probability. People then say the probability means all sort of things that make no sense. For example, some say the probability of getting an outcome is the proportion of that outcome in an infinite sequence of measurements. No such sequences actually exist. Also you could change the proportion of that outcome in an infinite sequence just by changing the order of the sequence. For example the infinite sequence that consists of repeating 01 an infinite number of times has the same number of 1s and 0s as the sequence 001 repeated an infinite number of times, so all that has changed is the order in which the 1s and 0s occur. So if you were to take 010101… as giving a probability of 1/2 for 1 and 001001… as giving a probability of 1/3 for 1 then you could change the probability just by changing the order. There are other bad ideas that I won’t go into here. In quantum mechanics, the probability of an outcome is a way of counting the instances of that outcome that tells you how you could bet on them in such a way that other people could not construct a strategy to make you lose consistently.

If your system is in the state $|0\rangle$ and you measure B, then in one universe you see + and in another you see -. The probability of seeing + is 1/2 and the same is true for -. the way you get the probability for + is by looking at the state $|0\rangle$ and squaring the magnitude of the number in front of the state $|+\rangle$ in that state. The probability for – is obtained by the same procedure. There may be other physical quantities you could measure for your system and they would have different sets of possible outcomes and different probabilities. If you wrote down $|0\rangle$ in terms of any other set of states representing possible outcomes of a possible measurement on that system, then you would use the same rule to get the probabilities for the results of that measurement: square the magnitude in front of the relevant state. All of the predictions it is possible to make about a system are contained in the set of possible outcomes for each quantity and the probabilities of those outcomes. So if two instances of a system had the same set of possible outcomes and probabilities for every physical quantity that could be measured there would be no way for you to tell them apart: they would be physically identical in every way.

Now, from rule (1) you could write
$|0\rangle= \tfrac{1}{2}(|0\rangle+|0\rangle),$
or
$|0\rangle= \tfrac{1}{3}|0\rangle+\tfrac{2}{3}|0\rangle,$
or
$|0\rangle= \tfrac{1}{\pi}|0\rangle+(1-\tfrac{1}{\pi})|0\rangle.$
So the instances of the system with the value 0 can be divided up into sets in a continuous number of different ways. In all of those divisions, each set has the state $|0\rangle$, and makes the same predictions about the sets of possible outcomes of measurements and the probabilities of those outcomes for each outcome. So although there is more than one instance of the system in which A has the value 0, they all make exactly the same predictions about the values of all measurable quantities. There is more than one instance of 0, but they are physically identical: they are fungible.