Why quantum theory isn’t a shell game (PBR Theorem for the layperson)

The following is an excerpt from an Appendix of my forthcoming book, Understanding Our Unseen Reality: Resolving Quantum Riddles (Imperial College Press). In the book, quantum states (‘kets’) are represented by triangles.

The proof of Pusey, Barrett and Rudolph demonstrates that quantum states cannot be taken as approximate descriptions of underlying, hidden properties. To present this simplified version of the proof, we’ll be making use of electron spin. An electron that was measured in a Stern-Gerlach apparatus with its magnet oriented in the vertical direction could be found to be spinning either ‘up’ (U) or ‘down’ (D). Suppose one of these electrons was found ‘up.’ It would be described by the appropriate possibility triangle:

Let’s represent this labeled triangle by the notation “|U>”. Now imagine that we have another S-G box with its magnet oriented horizontally and to the right, so that an electron going through this device could end up spinning either ‘right’: |R> (‘up’ with respect to the horizontal direction) , or ‘left’: |L> (‘down’ with respect to the horizontal direction). It so happens that the possibility |U> contains equal amounts of |R> and |L>. This means that the electron is in a superposition of equal amounts of |L> and |R>:If it is then detected at the leftward detector, this means (in the usual way of understanding the theory, which neglects the absorption process) that its state has ‘collapsed’ to the possibility triangle labeled by “L”:

Here is where we encounter the mysterious ‘collapse of the wave function’ that is inexplicable under the usual ways of approaching quantum theory: if the collapse is something that occurs in spacetime, it is clearly in conflict with relativity. This is the problem that is seemingly evaded by using the “Knowledge Interpretation” (KI) of the quantum state. The KI says that the collapse was not something that happened to the quantum system – it was just our knowledge becoming more precise and focused. According to the KI approach, that electron has some real physical property – let’s use Einstein’s term for this, an element of reality–that we can’t readily get at. But every time we measure it, the ‘collapse’ of the state just represents our knowledge becoming more precise — narrowing in on that hidden element of reality. Let’s call the hidden property EOR for ‘element of reality.’[1]

The idea is that the electron really had property EOR all along, but it could be labeled either by |U> or by |L>, with some probability for each. That is, having the property EOR would predispose an electron to be found in |U> with some probability and to be found in |L> with some other probability. It’s important to note that these probabilities are not part of standard quantum theory, which does not acknowledge anything like the EOR. They arise only in the Knowledge Interpretation, in order to connect its proposed EOR to the usual quantum state. So to keep track of this point, let’s call these EOR-probabilities. To see how this kind of ‘knowledge interpretaton’ makes sense in ordinary life, consider the following ‘hardware store’ analogy.

Suppose you need a home repair item, say a new handle for a cabinet, and you decide to go to the new hardware store in town to get one. We could label your initial state of knowledge about the handle’s location with the symbol |S>, representing the entire 100,000 square foot store. Now you make your first ‘measurement’: you walk into the store and see a sign, “Hardware”, hanging at the far right of the store. As you approach, you see that the hardware section occupies about ¼ of the store area. Your state of knowledge of the item’s location has just ‘collapsed’ to |H>, which represents a specific 25,000 square foot region. Now you make another ‘measurement’: you ask an employee where the cabinet handles are, and he points you to Aisle 12. Once again, your state of knowledge ‘collapses’ to the more focused, precise state of |A> which represents only that aisle. Finally, as you walk down the aisle, you spot the handle you are looking for. Your knowledge state finally ‘collapses’ to |X> – the precise spot where the handle is sitting on a shelf. Each of these collapses, to S, H, A, and X, pertain strictly to your subjective knowledge; none of them is an objective, physical collapse.

Here’s why the above approach might seem to allow us to preserve a traditional, classical understanding of reality in which it is assumed that for a process to be physically real, it must occur within spacetime. If ‘collapse’ is a physical process occurring within spacetime, then it appears to be in conflict with Einstein’s theory of relativity. This is because the ‘collapse’ seems to necessarily propagate at infinite speed, while relativity requires that no physical signal can exceed the speed of light.[2] But in the description above, all of these ‘state collapses’ were about your knowledge, not about the handle — it had its position all along, you just didn’t know it. Thus, many researchers have found this an appealing way to try to understand the ‘collapse of the wave function’: perhaps the wave function describes our knowledge about a system, but not the real, hidden facts about the system. So using a “Knowledge Interpretation” of the quantum state – just as in the above ‘knowledge interpretation’ concerning the hardware item, might seem like a good alternative to taking ‘collapse’ as a real physical process.

However, it turns out that quantum mechanics itself will not allow us this ‘Knowledge’ interpretation of the quantum states, as appealing as it may seem to some. This was shown by the team of Pusey, Barrett and Rudolph in 2011. In a nutshell, using the assumption of an EOR ‘inhabiting’ quantum states turns out to conflict with the well-confirmed predictions of quantum theory.

To set up the PBR argument, imagine the hidden property EOR as a ‘masked man’ that is hidden and inaccessible to measurement, but which secretly inhabits more than one possible quantum state, in this case both |U> and |L>:

In the knowledge interpretation illustrated above, the physically real property of the system, EOR, is in the overlap between the two states |U> and |L> containing the ‘masked man’ (which represents EOR), while the empty areas of the states represent our ignorance of the actual physical situation. It’s important to understand that EOR is neither |U> nor | L> ; it is not a quantum state, just some alleged hidden property that is not directly detectable. In terms of our ‘hardware store’ analogy, the hidden property EOR is like the cabinet handle that we haven’t found yet, and the quantum state descriptions such as |U> or |L> are like the different store areas (‘hardware’ or ‘carpentry supplies’) where it might actually be displayed. Using these basic concepts, the PBR theorem will show that there can be no such ‘hidden’ real feature of our quantum system; the ‘collapse’ cannot be interpreted as just our ignorance shrinking and our knowledge becoming more precise. In order to show this, we’ll need to consider two electrons subject to the measurements discussed above, as well as some additional well-established features of quantum theory.

Remember that one S-G device measured the ‘vertical’ aspect of the electron’s spin (which could be either U or D), while the other one measured the ‘horizontal’ aspect (which could be either L or R). In order to do the proof, we need to use a system of two electrons, each of which could theoretically be prepared in either U or L. Let’s represent these 2-electron states by very long triangles, with patterns indicating the property U or L. The four 2-electron possibilities look like:

The top left-hand triangle is the state for which both of the electrons are ‘up’; the top right-hand triangle is the state for which electron 1 is ‘up’ and electron 2 is ‘left’; the lower left-hand triangle is the state for which electron 1 is ‘left’ and electron 2 is ‘up’; and the lower right-hand triangle is the state for which both electrons are ‘left’. (You can also think of each of these tall triangles as two single-electron triangles for the relevant states ‘stuck together,’ but for our purposes below, it’s better to represent the state of the two electrons as a single triangle representing both of the particles.)

Now, suppose that (unbeknownst to us) both electrons actually do possess the hidden state EOR, but we haven’t yet done the S-G measurements on them to find out whether each of them is “U” or “L”. (Each electron is measured by its own S-G device, which can be set to measure either the vertical or horizontal direction.) According to KI, the various possibilities for the two-electron states must look like this (we omit the patterns in order to see the EORs clearly):

That is, all the tall ‘knowledge state’ triangles must contain the hidden real states, because according to the KI, there has to be some overlapping area — common to all these states — that contains the hidden properties actually possessed by each of the electrons. This is a crucial point, because it is what is meant by a ‘knowledge interpretation’: the idea that a quantum system really has some property and that the quantum state is just an approximate description that can be sharpened based on new information. This means that different approximate descriptions (UU,..LL) can apply to the same hidden property, in the same way that there are two approximate, overlapping ways to describe the location of our cabinet handle in the hardware store (either by ‘hardware’ or by ‘carpentry supplies’). According to KI. these approximate triangle state descriptions are all we get when we do a quantum measurement – we cannot get at the exact element of reality, even if it is there.

For the next step in the proof, we need to consider an additional, different kind of measurement that we can make on the set of both electrons together. The states representing the possible outcomes of this next measurement are a bit more complicated to write down, but we don’t need their explicit forms for purposes of our discussion. The four possible outcomes of this particular measurement are basically the opposite properties of each of the above states. Let’s therefore label those four possible outcome states as follows:

It’s important to note that whenever we do this second type of measurement, we must get one of these outcomes. And here’s where quantum theory itself rains heavily on the KI parade. As might be obvious from the names of the states, the theory tells us the following when we perform this measurement on the two electrons:

• If the two electrons were in state |UU>, one would never get |NOT UU>.
• If the two electrons were in state |UL>, one would never get  | NOT UL>.
• If the two electrons were in state |LU>, one would never get  | NOT LU>.
• If the two electrons were in state |LL>, one would never get  | NOT LL>.

This makes logical sense: if the two electrons are found to be in a certain state, regardless of what that state is, a measurement performed on them should never yield a result that directly contradicts that state. In fact this is what we find in the ordinary S-G measurement: if we input the state |U> into an S-G device oriented in that same direction, we never get the opposite result, |D>.

So suppose both electrons happen to have some hidden state EOR, and they are first found in the quantum state |UU>, which means ‘both electrons are up’.  Quantum mechanics (and basic logic) demands that if the two electrons were in this state and the second kind of measurement were then performed on them, they could not be found in the state |NOT UU>. This is because that state means precisely the opposite of |UU>; i.e., that the electrons are not both ‘up’. So to be consistent with that, we have to say that when both electrons have the hidden property EOR they cannot have any chance of yielding the outcome |NOT UU> when this second kind of measurement is performed. This is because there is a possibility that when both electrons have the hidden property EOR, they might end up in the state |UU>. But if two electrons are in this state and then the second kind of measurement is performed, we must never be able to get the outcome |NOT UU>, which would contradict the state |UU>. So, we must conclude that a set of two electrons both possessing the hidden property EOR must have zero EOR-probability of ending up the state |NOT UU> when the second kind of measurement is performed. This is because there is a chance that they might be found in the state |UU>, and if in that state, they can never be found in the state |NOTUU>.

Here’s a way to see this in terms of the hardware store analogy: again, the EOR is represented by the cabinet handle. In this store, there is a hardware section and a gardening section, and there are never any items in common between those two sections. Now suppose the handle could be stocked in home improvement. Then clearly it can never be found in the gardening section. So we must simply say that our handle has a zero EOR-probability of ever being found in the gardening section of the store.

But the same argument holds for all the other possible outcomes for the first kind of measurement (represented by the other three tall striped triangles pictured above). For example, the hidden state could still be “both electrons possess EOR” and the outcome found from doing the first kind of measurement could turn out to be |UL>. For consistency, the opposite outcome “NOT UL” cannot occur when the second kind of measurement is performed, so the EOR-probability for that outcome must also be zero. And so on, for all the possible outcomes of this second measurement. But this is absurd, because it leaves us with the following situation: given a real property EOR possessed by both electrons, and a legitimate measurement that could be performed on them (the second measurement we discussed), there is zero probability for any of the outcomes of that measurement to occur. This is nonsense, because we know that when we do the measurement, we will certainly get one of those outcomes.

Finally, here’s a simpler way to visualize the gist of the proof: consider a shell game (pictured to the left), in which a magician places a pea under one of a set of four white shells. Then he quickly moves the shells around so that we don’t know which shell contains the pea. In this parable, the pea is the EOR and the white shells are the four possible quantum states that we get as outcomes when we perform the first kind of measurement. But this magician can’t cheat – he cannot remove the pea (just as, according to the key premise of KI, the quantum system really does have some hidden property EOR). Now suppose the magician swaps each of the original white shells with a black shell, in such a way that we still can’t see where the pea ended up. It still has to be there, doesn’t it? But, according to the proof, the pea has vanished. It cannot be under any of the black shells, or we will either contradict quantum theory or our own premises about quantum systems having a hidden property that could be found under any of the original white shells.

The problem here is caused by the idea that a real, but hidden, property EOR could result in more than one distinct quantum state, in this case both |U> and |L> . In terms of our shell game analogy, the problem is the idea that there is a pea that could be under any one of the shells – where the shells represent the quantum states. Remember that EOR is not a quantum state, but some property that supposedly could be common to two different quantum states, just like a cabinet handle could be classified as either ‘hardware’ or ‘carpentry supplies’. This is what is meant by the idea that the ‘collapse’ of the quantum state describes not a real physical collapse but rather a sharpening of our knowledge about a quantum system. It implies that there is more than one possible state of knowledge about the same hidden truth – just as our knowledge of the cabinet handle’s true location could be approximately described in different ways, or the pea might be found in any of the shells. The proof shows that quantum mechanics is not consistent with this common sense idea, since it leads to a logical absurdity. We therefore must conclude that the quantum state does not describe our knowledge in the sense captured by the proof– it describes something with an indivisible uniqueness.[3] In terms of the shell game, there is no hidden pea. The “shells” – the quantum states themselves — are the most precise descriptions of the true reality of quantum systems.

Notes:

[1] Any resemblance of this term to a Winnie-the-Pooh character is unintentional, but interesting nevertheless.

[2] The reason that collapse seems to be instantaneous in the standard approach to the theory is because absorption is neglected. This leaves only the offer wave to mysteriously collapse all by itself as seen by the observer who is measuring it. Researchers who have studied the collapse under these circumstances have concluded that it has to occur along a ‘line of simultaneity’ in a particular frame. Readers interested in this issue, and how the transactional process remedies the problem, can learn more by reading Kastner (2012, §6.7)

[3] Some proponents of ‘knowledge’ interpretations argue that it is possible to retain the KI by exploring loopholes in the proof. But this undermines the main motivation for KI as a commonsense way of avoiding collapse, since the loopholes involve stranger notions of what the underlying reality might be. One therefore faces a kind of ‘diminishing returns’ situation with this approach.