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Tumbling Down a Quantum Rabbit Hole

Probability does not exist. Quantum mechanics is personal. Confused? So was I when I tumbled down the rabbit hole called QBism three years ago. I really wasn’t looking to have my worldview shaken that day. With less than a year left to finish my PhD in particle physics, I had more pressing concerns than the weird philosophical conundrums lurking at the bottom of physics. I was only searching for some background material on statistical methods when I found a document by quantum physicist Christopher Fuchs called My Struggles With The Block Universe.

Intriguing title, I thought, hit the print button and left for lunch. Only afterwards did I realize that I had asked the poor printer to spit out a 2349-page collection of Fuchs’ emails. This collection is Fuchs' way of documenting ten years of discussions with numerous colleagues on one of the strangest questions in physics: how should the theory of quantum mechanics be interpreted? At centre stage is a view developed mainly by Fuchs and Ruediger Schack called Quantum Bayesianism, or QBism. Now you may wonder what Thomas Bayes, an 18th-century Presbyterian minister, could possibly have to do with quantum mechanics but don’t worry, we will get to that in due time.

Publishing one’s own emails at first struck me as somewhat big-headed. But following the advice in Maximilian Schlosshauer’s foreword to My Struggles With The Block Universe, I decided to flip my way to a random page and read on for a bit. Three years later I still haven't crawled back up from the QBist rabbit hole. In fact, part of me is starting to rather like it down here.

QBism is only one of many proposed interpretations of quantum mechanics, all of which are rather weird, but in fundamentally different ways. (It is perhaps telling that a recent classification of the main interpretations was titled Interpretations of Quantum Theory: A Map of Madness.) Yet, there is something about QBism that strikes me as particularly interesting and beautiful. I hope to convey to you why QBism fascinates me—and why I'm not completely sold on it yet. But before we get to QBism we must understand how questions of interpretation arise in quantum mechanics in the first place, and what role our understanding of probability plays in all this.

We don’t understand quantum mechanics

Quantum mechanics is the mathematical theory that physicists use when dealing with nature at the very small scale of atoms and below. But more than 90 years since it was first formulated by Werner Heisenberg and Erwin Schrödinger, there is still no consensus on what this theory really is telling us!  Last year, physicist Hans Christian von Baeyer published the first popular science book on QBism. He points out that one likely source of confusion is that quantum mechanics represented a break from the pictorial descriptions that dominated much of pre-quantum, or classical, physics.

The world used to be a lot simpler. Starting from intuitive, mechanistic models based on simple concepts such as small pieces of matter and the forces between them, physicists were able to find mathematical descriptions for an impressive range of phenomena. We can illustrate how this works by considering a standard example from physics textbooks, namely the wavy motion of a guitar string. To derive a mathematical description of the guitar string, you start with a simple sketch of how neighbouring pieces of the string pull on each other. Then you add Newton's laws, stir around using the elegant maths of Newton and Leibniz and—voilá—out pops a beautiful wave equation. Now, since the starting point was a simple picture of how each little string piece behaves, the question of how the final equation should be interpreted never arises. It is a description of the string's movement, end of story. This straightforward link between the initial mechanistic picture on the one hand and the mathematics on the other is typical for classical physics.

Things took a turn for the weirder with the birth of quantum mechanics. During the early twentieth century it became clear that mechanistic models were unable to capture nature's behaviour at the atomic level. To find equations that matched the strange results of atomic experiments, the clarity of classical physics had to be surrendered. The equation that lies at the heart of Schrödinger's formulation of quantum mechanics can be seen as a kind of wave equation. But there is a crucial difference to the example of the guitar string: in Schrödinger's equation it is far from clear what is waving! The "string" in this case is a mathematical object known as the wavefunction (or equivalently, the quantum state). The wavefunction is the “main character” in the equations of quantum mechanics—but, rather embarrassingly, we still don't know exactly what it represents.

At first glance it may seem like Max Born settled this question already in 1926. Born realized that the wavefunction was intimately connected to probability. Loosly speaking, the Born rule says that if you multiply the wavefunction with itself you get a probability. In other words, knowing the wavefunction will typically not let you predict the detailed outcome of an experiment—like the exact position where a particle will hit the detector of the experiment, or the exact time that a radioactive atom will decay. But it does allow you calculate the probabilities for all the possible different outcomes. This is the famous indeterminism of quantum mechanics.

The Born rule provides the main link between the mathematics of quantum mechanics and the results of physical experiments. Physicists who prefer to keep philosophical questions at arm’s length can just apply the Born rule when needed and get on with their work. But if we are to try to understand what quantum physics really is about, we’ll have to dig a little deeper. For while there is little doubt that the Born rule gives a correct relation between the wavefunction and probability, it does not tell us what type of probability we are dealing with. As we will see, this question can have profound implications for how we understand quantum mechanics.

Actually, we don’t understand probabilities either

Like quantum mechanics, probability theory also suffers from some interpretational confusion. In fact, the Stanford Encyclopedia of Philosophy identifies no less than six main approaches to interpreting probability. Perhaps there is no single interpretation that can encompass all aspects and uses of probability. Yet, it seems reasonable to expect that at least those probabilities appearing in a well-defined mathematical theory like quantum mechanics should lend themselves to unambiguous interpretation.

The probability interpretation that is commonly taught in school is frequentism. As the name suggests, frequentism identifies probability with relative frequency, that is the fraction of trials that yield a given outcome in an experiment that is repeated a large (hypothetically infinite) number of times. (Think repeated coin tosses or dice rolls.) As Baeyer comments in his book, this connection to experiments ensures that frequentist probability “assumes an air of objectivity,” something that may help explain why the frequentist viewpoint, at least traditionally, has been dominant among scientists.

Among philosophers, however, frequentism seems less popular. To get a feeling for just one of the many problems that arise in the frequentist interpretation, consider a simple coin toss experiment: say you want to measure the frequentist probability for heads—which I shall refer to as P(h)—with a given coin. To determine the “true” P(h) you would have to toss the coin infinitely many times—which is clearly impossible. In the real world, you can only do a finite number of repetitions and use your collected data to make an estimate with some uncertainty, such as P(h) = 52% ± 3%. But for this probability to make sense, the uncertainty also needs a frequentist interpretation. This will then refer to another impossible infinity of repetitions, this time of the entire experiment. But this can again only be approximated by finitely many repetitions, leading to an “uncertainty on the uncertainty”—and so on and so on. Thus, in the real world of finite experiments, the frequenstist understanding of probability ends up chasing its own tail in what looks suspiciously like circular reasoning.

Richard Feynman, who among other things invented the iconic squiggly diagrams that particle physicists like myself use when calculating probabilities, comments on the problem of probability interpretation in his legendary Feynman Lectures on Physics (1964). After introducing probability in a mostly hands-on, frequentist fashion, he warns the reader against thinking that there exists some “true” probabilities out there, ready to be experimentally measured by physicists:

There is [...] no way to make such thinking logically consistent. It is probably better to realize that the probability concept is in a sense subjective, that it is always based on uncertain knowledge, and that its quantitative evaluation is subject to change as we obtain more information.

Feynman touches here on a different class of probability interpretation in which a probability represents a degree of belief. This is called Bayesian probability, originating with the work of 18th-century theologian and mathematician Thomas Bayes, and further developed by Pierre-Simon Laplace. In contrast to frequentism, which relies on the concept of repeatable trials, Bayesian probability can in principle be applied to any proposition, such as “QBism is correct” or “the moon is made of cheddar.”

Broadly speaking, a Bayesian degree of belief has two possible interpretations: either as an objective measure of rational support for a proposition, or as a subjective measure of an individual’s belief in a proposition. One of the founders of the subjective Bayesian viewpoint was the Italian statistician Bruno de Finetti. In the preface to his book Theory of Probability (1974) he writes:

My thesis, paradoxically, and a little provocatively, but nonetheless genuinely, is simply this:


The abandonment of superstitious beliefs about the existence of Phlogiston, the Cosmic Ether, Absolute Space and Time, ... , or Fairies and Witches, was an essential step along the road to scientific thinking. Probability, too, if regarded as something endowed with some kind of objective existence, is no less a misleading misconception, an illusory attempt to exteriorize or materialize our true probabilistic beliefs.

This is the starting point for the QBism of Fuchs et al: to assume de Finetti's personalist view of probability and explore what consequences this must have for the interpretation of quantum mechanics.

Quantum mechanics according to QBism

To summarise the story so far: Quantum mechanics is a (spectacularly successful) theory of fundamental physics that allows us to make probabilistic predictions, but it doesn’t tell us precisely how these probabilities should be interpreted. Turning to the philosophy of probability, we find that there are a handful of competing probability interpretations. One of these is subjective Bayesianism, which views probability simply as a way to quantify personal degrees of belief.

This brings us finally back to QBism. Simply put, QBism is the combination of quantum mechanics and subjective Bayesianism. This merger is motivated both by philosophical arguments for Bayesianism, and, as we will see later, its potential for dissolving some of the notorious mysteries of quantum mechanics. But before we get to that we must look at precisely how subjective Bayesianism shapes the QBist view of quantum mechanics.

Even in quantum physics new ideas rarely pop out of vacuum. QBism has a lot in common with the widely adopted (but notoriously hard to define) Copenhagen interpretation of quantum mechanics, originating in the works of Niels Bohr and Werner Heisenberg in Copenhagen in the late 1920s. Despite being referred to as a single interpretation, it is rare to find two Copenhageners that understand quantum mechanics in exactly the same way—this was not even the case with Bohr and Heisenberg. However, a common feature of Copenhagen-like interpretations is that the wavefunction is regarded not as something physically real, but as a mathematical tool that encodes all that can be known about a quantum system.

But now we can go one step further and ask “Who is doing the knowing here?” In other words, whose wavefunction is it? The QBist answer to this question is aptly summarised in the insisting lyrics of the late Soundgarden singer Chris Cornell: “My wave! My wave! My wave! My wave!” This is one of the points where QBism makes a crucial departure from the Copenhagen interpretation. In QBism all probabilities express an agent’s personal degrees of belief. Since the wavefunction encodes probabilities, the conclusion is that the wavefunction itself must be personal to the agent!

Few people thought as much about the meaning of quantum mechanics as Bohr. In 1929 he wrote:

In our description of nature the purpose is not to disclose the real essence of the phenomena but only to track down, so far as it is possible, relations between the manifold aspects of our experience.

The QBist view is similar, with one important clarification: if probabilities are personal, then so too are the possible experiences of which they speak. Thus, quantum mechanics is for each of us a theory about our own probabilities regarding our own possible experiences when interacting with the indeterministic quantum world.

It’s not all in your head

Solipsism—the idea that the external world is nothing but an illusion created in your mind—is understandably not very popular among scientists. But with all its talk about personal probabilities and experiences, isn’t QBist quantum mechanics just solipsism in disguise? This seems to be the concern for most physicists when they first encounter QBism, as it was for me. But the concern is misplaced. QBism raises no doubts about the existence of the world external to the agent. It is quite the opposite, actually. The fact that the agent is forced to navigate the quantum world using probabilities—not certainty—is an expression of the independent existence of the external world. The agent may choose to kick the world in a specific way, but she can never be sure exactly how the world is going to kick back.

And even if quantum probabilities and experiences are personal, that does not mean quantum mechanics cannot provide objective insight about the physical world. When historians study the medieval Norwegian text Konungs skuggsjá (The King’s Mirror), they don’t read the text as a factual account of a conversation between a father and son. They know that the dialogue belongs to the very specific genre of speculum literature—educational texts for future kings (in this case: King Magnus Haakonsson)—and should be interpreted accordingly. With this in mind, historians can subject Konungs skuggsjá to all kinds of analyses in order to extract objective knowledge about the medieval world.

QBism makes us approach quantum mechanics in an analogous way. To extract objective knowledge about the physical world from the mathematics of quantum mechanics, we first need to understand what “genre” quantum mechanics is written in. And according to QBism, the genre of quantum mechanics is that of a user’s manual. It instructs an agent how she should relate and update her various expectations when dealing with the quantum world.

This manual-picture helps QBism disentangle the subjective and objective aspects of quantum mechanics. An agent’s expectations, formulated in the language of wavefunctions, may well be subjective, but the underlying structure of quantum mechanics is certainly not. In QBism, equations like Schrödinger's equation and the Born rule represent rules of consistency for any agent’s expectations concerning the quantum world, and as such, they are deeply connected to objective aspects of nature. As Fuchs writes, “if quantum theory is a user’s manual, one cannot forget that the world is its author.”

Collapsing the mysteries

Now, this next section is somewhat technical. If you have reached your threshold for silly-sounding words and weird concepts, I suggest you skip to the last section (“A subjective doubt of subjectivity”). However, if you are interested in how the subjective viewpoint of QBism tackles the quantum mysteries of wavefunction collapse and entanglement, do read on.

At the technical level, the central issue that has lead to the wide variety of quantum interpretations is the problem of wavefunction collapse. This is the question of what happens when we perform a measurement on a quantum system. For this reason it is also known as the measurement problem. The story goes something like this: prior to the measurement the Schrödinger equation tells us that the wavefunction just waves along in a nice, smooth fashion. At this point the wavefunction will have some “spread-out” shape that encodes probabilities for all the potential outcomes of the measurement we are about to perform. But at the very moment of the measurement something interesting happens: there is seemingly an abrupt change in the shape of the wavefunction, from its previous spread-out shape to a narrow peak that singles out the one outcome we actually observe.

In a quantum measurement, the wave function, and thus the associated probability distribution, instantaneously collapses to a shape that singles out the observed outcome.

In a quantum measurement, the wave function, and thus the associated probability distribution, instantaneously collapses to a shape that singles out the observed outcome.

This sudden “collapse” of the wavefunction is not described by the Schrödinger equation, or any other standard equation in quantum mechanics, so how are we to understand it? The various interpretations of quantum mechanics present very different answers to this question. In QBism, wavefunction collapse happens to be one of the least mysterious aspects of quantum mechanics. Since the wavefunction encodes the beliefs of an agent, the collapse simply reflects the sudden change of information experienced by the agent upon learning the measurement outcome. This is nothing different from what happens the moment you learn about the outcome of a lottery: Say the lottery consists of drawing a single number from 1 to 100. Before you know the result you are in a state of uncertainty, so you may reasonably choose to assign a 1% probability—that is, a degree of belief of 1%—to each number from 1 to 100. But as soon as you see the drawn number you move to a state of certainty. Suddenly you believe 100% in the observed outcome, leaving 0% belief for all other outcomes. Your previously spread-out state of belief has collapsed onto the single outcome you experienced. 

However, the QBist story of quantum measurements is not only about updates in beliefs. The measurement situation consists of two parts: the agent and the external quantum system. Any quantum measurement represents some action by the agent upon the system, and this action unavoidably has some impact on the world. Fuchs often points to John Wheeler’s formulation that “each elementary quantum phenomenon is an elementary act of ‘fact creation.’" In QBism, a quantum measurement is seen as the special case where an agent takes part in this fact creation process, and experiences the outcome of it.

Wavefunction collapse also plays a key role in the description of quantum entanglement. This phenomenon is often presented as proof that quantum mechanics must allow for some kind of instantaneous interaction across arbitrarily large distances—known in physics as non-locality—but this is misleading. Non-local interactions are only necessary within some interpretations of quantum mechanics. QBism is one of several interpretations that avoid invoking such “spooky action at a distance,” as Einstein famously called it. However, what is rather unique about QBism is the way in which this is achieved.

Say you have a heavy, unstable particle that decays into a pair of lighter particles, A and B. The two particles fly off in opposite directions so that they are soon separated by a large distance. While you perform a measurement with particle A over here, a colleague will perform a measurement with particle B over there. As the particle pair originated in the same physical event, the combined system of A and B is expected to exhibit certain correlations, which can be expressed mathematically in the initial wavefunction for the system. When you then carry out your measurement with particle A, the wavefunction for the combined system of A and B collapses. The collapsed wavefunction leaves you with a given probabilistic prediction for a measurement with particle B. However, if you had chosen to do a slightly different measurement with A, the wavefunction would have collapsed in a different way, resulting in a different probabilistic prediction for particle B! Does this mean that you can exert instantaneous, non-local influence on the far-away particle B by choosing a measurement and conducting it with particle A?

QBism says no. Your wavefunction, whether collapsed or not, only concerns your future experiences. It does not speak of the experiences of your far-away colleague, and it certainly does not tell particle B what it can or cannot do. The collapsed wavefunction simply tells you this: that in order to be consistent with the initial wavefunction you assumed for the system and your subsequent experience with particle A, you should now hold this particular set of beliefs about what you will experience once information about particle B reaches you.

Simply put, there is no room for strange non-locality in QBist quantum mechanics because all the experiences of an agent necessarily take place at the agent’s position in time and space. Again we can see how QBism takes to heart Bohr’s statement that the purpose of quantum mechanics “is not to disclose the real essence of the phenomena but only to track down [...] relations between the manifold aspects of our experience.”

A subjective doubt of subjectivity

I admit it, I find QBism very strange. Still, there seems to me to be a compelling honesty to the QBist approach. Since quantum mechanics is a theory of probabilities, we better make sure we understand what these probabilities truly mean. QBism starts by assuming what is arguably one of the simplest probability interpretations around, subjective Bayesianism, and then explores how this impacts our understanding of quantum mechanics.

Much of the murkiness of quantum interpretation seems related to the divide between the subjective and the objective. Schrödinger, uncomfortable with the implications of the theory he had helped devise, put it this way in a letter to Arnold Sommerfeld in 1931:

One can only help oneself through something like the following emergency decree: Quantum mechanics forbids statements about what really exists—statements about the object. It deals only with the object-subject relation. Even though this holds, after all, for any description of nature, it evidently holds in quantum mechanics in a much more radical sense.

Through its commitment to subjective Bayesianism, QBism holds that probabilities in quantum mechanics are precisely about this “object-subject relation” and nothing else. It is this perspective that allows QBism to rather effortlessly navigate the treacherous waters of collapsing wavefunctions.

Still, I find myself not entirely convinced by QBism. Like many other physicists, I started with a strong, subjective belief that subjective beliefs should have no role to play in quantum mechanics. And even though I find the arguments put forward for QBism both elegant and rather compelling, they have yet to completely outweigh my initial distrust of subjectivity.

This philosophical prejudice may ultimately come from the historical development of physics. The main hints of a connection between physical theories and decision-making subjects are found in quantum mechanics. But before quantum mechanics, we developed successful theories for the macroscopic world of classical physics. Here the role of the subject was, at most, that of a passive observer. Given the success of these theories, it is not surprising that it seems so obviously reasonable to read physical theories as agent-independent descriptions of nature. After all, this is how we got used to doing physics in the first place. It’s hard to give up on old habit.

QBism is a work in progress. Perhaps the move to subjective probabilities will prove a huge leap forward in our understanding of quantum mechanics—or perhaps it is a step down a blind alley. But the mere possibility that something like QBism is correct is, at least for me, a forceful reminder of two related points.

First, that any attempt at connecting the equations of physics to statements about the nature of reality is fundamentally also a philosophical endeavour. Despite the impression one may sometimes get from (rightfully) proud physicists and advocates of science, physics alone does not hold the answers.

Second, that the true correspondence between the mathematics of quantum mechanics and our physical world may well be something rather different from what many of us are used to thinking. In particular, a consistent understanding of quantum physics does not have to imply the sort of simple reductionism that fundamental physics is often associated with. Christopher Fuchs puts it this way:

What is being realized through QBism’s peculiar way of looking at things is that physics actually can be done without any accompanying vicious abstractionism. You do physics as you have always done it, but you throw away the idea “everything is made of [Essence X]” before even starting. Physics—in the right mindset—is not about identifying the bricks with which nature is made, but about identifying what is common to the largest range of phenomena it can get its hands on. The idea is not difficult once one gets used to thinking in these terms.

It’s an intriguing idea. But I am far from used to it yet.

Selvets Tidsalder

Selvets Tidsalder

The Paradox of Flexibility

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