The Measurement Problem

Alright, let's talk about the measurement problem! This is defined by different thinkers in different ways, so it isn't always clear how we should talk about it.

This is how I have chosen to state the measurement problem, and it is admittedly arbitrary. Every interpretation chooses a different gripe to have about the measurement problem, and will likely formulate the problem in a way that makes their gripe, and subsequent solution, most obvious.

I don't aim to avoid bias in my formulation, I just want to define the problem in the biggest possible strokes that make sense to me. That is to say, I want a formulation of the measurement problem that most clearly frames the issues being debated and allows interpretations to be categorized and their relationships understood quickly.

The Measurement Problem

Some variables:
Let's call the observing mind of the scientist "O",
the macroscopic-sized measuring apparatus "A",
and the quantum state of the system being studied "Q".

The measurement problem could then be stated as three propositions which cannot all be true. One of these must be false.

1. Under the right conditions, A can be in the same quantum state Q as a microscopic-sized system; the size of the apparatus does not matter. Q can be a superposition state.
2. O directly perceives the world. This means that if it perceives the state of A, it perceives its state as exactly what it is.
3. A superposition state represents multiple realities, or a reality that is indeterminate between multiple realities. Minds cannot exist in multiple realities, so O cannot perceive or otherwise interact with A if it is in state Q.

This particular way of formulating the measurement problem has the strength of relative generality, but has the flaw of being fairly abstract and difficult to grasp without some explanation. So, let's look at each part more closely.

'A can be in state Q'

This is the least controversial of the three. Basically, it says that the laws of quantum mechanics are true in the scientific sense. It is the one claim that is a posteriori, meaning it can be verified empirically. Specifically, this means two things: 

a) Large macroscopic sized objects can follow quantum laws under the right conditions.

This is a point that could be brought under debate. Could it be that a relatively large-sized object like a cat actually cannot be brought to a quantum state so that it would behave as quantum mechanics would predict? Or is there some sort of physical effect that prevents that from happening? Some interpretations argue that there is an effect like this.

b) Quantum states can be superpositions.

This is a fundamental part of quantum mechanics. It would be difficult for an interpretation to deny this. Since superposition states are so effective at explaining quantum phenomena, denying this would mean completely denying that quantum mechanics is true in even an approximate sense and that we need to start from scratch.

Accepting this statement does not mean that one believes quantum superposition states must be real things that exist in the world, which is a much more controversial statement. Nor does it mean that a quantum state must be interpreted as a literal physical state. Many interpretations would argue that quantum states should not be considered real physical states because there is some better notion of state out there that we have not yet discovered. These interpretations would still have to admit that the notion of quantum states and the notion that those quantum states can be in a superposition state, however those two things are to be understood, are essential in empirically successful quantum mechanics.

Like I said, this statement is the least controversial.

'O directly perceives the world'

This is a metaphysical statement. Essentially, this statement boils down to whether or not we perceive reality as it really is, or if reality is there is an aspect of reality we fundamentally cannot perceive.

The famous philosopher Immanual Kant introduced the notion of transcendent that is used in modern philosophy. It refers to something that is essentially beyond human experience. Does an interpretation allow for a transcendent reality? If so, that interpretation would deny this statement.

This is important to include because it establishes whether or not it would allow for there to be some kind of "hidden reality" of the metaphysical kind. For example, the multi-universe interpretation allows for possible worlds that are beyond our perception because we are trapped in our own possible universe.

I would think that many would find no problem in denying this proposition. Some interpretations accept this statement, and some don't, and it is a good proposition to use to divide and begin to categorize interpretations.

'Superpositions represent multiple realities'

This is where the problem comes into full form. Superpositions are the linear combination of two other quantum states. That is to say, add two quantum states and you get a superposition. Since superpositions are just this kind of mathematical combination of two other states, it seems natural to assume that they should be interpreted as representing both states in some way. 

Perhaps it is an indeterminate state that ceases to become one reality until collapsed. Perhaps it is a dual existence, one that represents both realities existing at the same time until it is required to be one or the other. Whatever the case, this statement says that it cannot represent just one determinate state.

And because of this dual existence, this is a problem for how we should understand what perception of a superposition should look like. Excluding weird and unhelpful ideas, we don't typically think of our minds as existing in multiple realities. Since this is the case, these extra realities have to be shed somewhere between the physical superposition and our mind's perception of it.

Many would lump the notion of superpositions representing multiple realities in with the baggage associated with quantum mechanics. I don't do this because it is useful to separate this a priori claim from the a posteriori claims of the first statement. It's a statement about how to understand the significance of the superposed quantum state, not a matter of the predictions made using it, though the latter can inform the former.

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So that is how I like to phrase the measurement problem. It is simple enough to grasp, while still showcasing many of the concerns interpretations try to address.

Quantum Superposition Paradox

Superposition is a weird thing, and there's definitely disagreement over how to think about it. Mostly, there is disagreement over whether or not it literally represents multiple realities. That is to say, when Schrödinger's cat is in a superposition, is it both alive and dead at the same time?

It's hard to talk about that without getting into metaphysical debates (see my entry on the measurement problem). While metaphysics is good to get into, it'd be nice to avoid that, at least initially.

So maybe a good starting place for talking about the interpretation of superpositions are logical sentences about them.

Multiple Realities

For example, if it's true that 'Schrödinger's cat is in a superposition state', does that logically imply the statement 'the cat is alive' as well as the statement 'the cat is dead'. More symbolically, is it the case that

'Superposition' -> ('alive' & 'dead') ?

I'd wager that this view of superpositions is widely held. Most will probably think of superpositions as implying multiple realities in the very meaning of the term. This would be an a priori assertion, and it would agree with the phrasing above.

Opposed to this view would be the belief that there is no true logical relation that looks like the one above. One could argue, for example, that superpositions could possibly imply multiple realities, but that it would have to be decided empirically. This would mean that the above logical statement wouldn't always be true, and that it would depend on the circumstances.

See how much cleaner that is than normal? I don't have to talk about metaphysical things, but still get to articulate something providing some insight.

Contradictory Realities

The second half of this is what relationship is between the states of life and death. Do they contradict? If so, the statement

~('alive' & 'dead')

is directly contradictory with the superpositions statement above. In other words, one of these two statements must be false.

This second statement certainly seems natural to accept as well. We certainly think of life and death as being mutually exclusive states. But what could show that they really are mutually exclusive states, and not just compatible states that appear incompatible to our mistaken intuition?

More formally, maybe we can say that mutually exclusive pairs of physical states will create two conflicting set of predictions, both of which can't be true. For example, a macroscopic object can be both falling and electrically charged. There is no reason why these two physical states should make conflicting predictions, so a physical object can be in both states at the same time.

(I think the full story of why this is the case looks like this: since a system is said to be in a physical state only when it fits a physical model, and physical models are what we need to make predictions, having a contradiction in the predictions of two models applied to the same system suggests an error somewhere further up in reasoning. Either a model was applied that should not have, or there was an error in how one or more of them were applied. Feynman describes using this very line of thought to call out the existence of errors in other physicist's reasoning and impress others without working through all of the math. It's in Surely You Must Be Joking Mr. Feynman p.244-5. The fact that the statement could be wrong either because an incorrect model was applied or a correct model was incorrectly applied means that this second statement has an a priori sense to it and an empirical sense to it.)

Two mutually exclusive states would give predictions that contradict. For example, having a macroscopic object (not currently under quantum effects) be both charged and not charged would create contradictions. There would be two sets of predictions created by the two possible situations: when the object is charged and when it is not, and those predictions could differ significantly. Therefore, only one of them can be true, and a well designed experiment will typically decide between the two.

So are quantum states different? Do quantum effects somehow make states that are normally mutually exclusive so that they are not in the right circumstances?

An interpretation that would argue that this statement is false is the many-worlds interpretation. While classical states like 'alive' and 'dead' are mutually exclusive, in the quantum realm they are not. The simple fact for this being that they exist in separate worlds. It's not a contradiction to say that the cat is both dead and alive because the dead cat exists separately from the alive cat, and it is this world-splitting aspect of quantum mechanics that keeps these two states from producing contradictory predictions.

So yeah, I dunno how helpful this approach will be, or if there is any literature on breaking it down this way. I'm calling it the quantum superposition paradox, which seems appropriate, but I don't know if there would be a better name for it.

Spekkens' 'Toy Theory'

I just read the paper about Spekkens' 'Toy Theory' of quantum mechanical phenomena. I had heard about this paper from the interview with Fuchs, where the paper is cited as being a support of the quantum bayesianism interpretation. I was not disappointed. This essay is awesome, and I think it is going to be seen as one of the most influential papers in in recent history. 

You should go read it here.

That's not to say that I think it is right in what it argues. Far from it, I think that the essay is fundamentally mistaken in many ways, though it is a smart and insightful fresh take on the subject. I think that there is much to learn from it, despite being at heart incorrect.

The paper's main goal is is argue that quantum states are epistemic and not ontic. Essentially, this is a claim against the realist interpretation of quantum mechanics. Ontic, here means that it is a theory which describes real objects in the world in a reliable way. The opposite of this is to claim that, though it enjoys predictive success, quantum mechanics at its base does not exactly describe the world. It is only a phenomenological theory, in a sense. It gives a scientist the means to predict how the world will behave, but it does so in a kind of incidental way. Phenomenological theories do not give causal accounts of why things behave the way they do, just a way to know what they will do. These types of theories are missing the causal mechanism that explains what happens.

Get it? Toy theory?

The alternative is epistemic, which is a version of non-realism that is plausible for quantum mechanics. Essentially, it claims that quantum states are not physical states, but are rather states of lack of information. Instead of encoding something about the world, the superposition of Schrödinger's cat just describes the information that we are lacking: the life or death state of the cat. Epistemic interpretations of quantum mechanics argue that quantum mechanics does not describe the real world, just a kind of lack of information about the world. Quantum Bayesianism is one of the more well known epistemic interpretations of quantum mechanics. 

This distinction between epistemic and ontic interpretations is a false dichotomy: it's not necessary that an interpretation fall into either of these categories. Furthermore, the dichotomy has a latent presupposition, but that is the subject of a later post.

Quantum states come in two different varieties: 'pure' and 'mixed' quantum states. A 'pure' q-state is essentially just a a quantum system described by one state vector in a superposition. 'Mixed' states are when a quantum system is described by multiple state vectors. A common view of many philosophers of science is that pure q-states are ontic, while mixed q-states are considered epistemic since they are just states of incomplete knowledge about what pure state a system is in. Spekkens argues that both pure and mixed q-states, however, are epistemic.

He claims that the epistemic view of q-states is superior to ontic view because certain phenomena like interference, noncommutivity, entanglement, no cloning, teleportation, etc. are mysterious in the ontic view seem natural in the epistemic view.

The success of this argument is a mixed one. On the one hand, this toy theory is useful for understanding some aspects of quantum phenomena. I think that it indeed is useful precisely because there is something true to it.

But I don't think Spekkens' claim is, strictly speaking, a fully true one. While it can be used for insight, the toy theory does not replace quantum mechanics in predictive power. This is why his argument is misleading: the toy theory cannot account for the list of phenomena above. For example, you cannot apply the toy theory alone to the situations which give rise to quantum interference phenomena. Quantum mechanics is capable of making empirical predictions, but the toy theory simply piggy-backs on those predictions.

This means that the toy theory is not a replacement of quantum mechanics in any way. This should be a red flag: scientific endeavors aim for predictive power. The 'toy theory' may be a strong philosophical aside, but it clearly requires another step backwards in understanding before we can go forwards.

Quantum-Intentional Interpretation

In my post on quantum mechanics and consciousness I mentioned a category of quantum interpretations that I call "quantum-intentional interpretations". I defined these as any interpretation that holds that consciousness can be a cause of physical quantum effects.

I want to revise this definition slightly, however. I was conflating two types of ideas. Exactly what aspect of consciousness is acting as a cause? There are two obvious ways of thinking of it:

1. Passive: conscious perception causes physical change.
2. Active: conscious decision of the way in which a quantum system is measured brings about a physical change.

(There may be more ways of thinking about this, but this seems exhaustive me.)

Obviously, the second is more appropriately called the quantum-intentional interpretation since it deals with the intentions of the quantum physicist. I want to use this as the new definition of the term. Interpretations that fall into first of the above distinction should be called quantum-observational interpretations.

There are a couple of reasons I accidentally conflated these two categories. The first is that my understanding of perception is that it needn't be understood as passive. That is to say, any passive description of perception could be phrased so as to be an active description. This is not an immediately obvious point, however. In addition, it's not clear whether or not this is actually true, much less whether or not I am justified in assuming it. Arguing this point would not be important for the discussion, so it is best to avoid implicitly assuming it.

There is a very good reason to make this distinction: one of these two interpretations is very easily refuted. It is easy to show that the quantum-intentional interpretation is plainly ridiculous.

The second is that the quantum-intentional interpretation was just so plainly false. But first, let's see what the most compelling argument in favor of it looks like.

Arguing in Favor of the Quantum-Intentional Interpretation

The technical term for the way a quantum state is measured is called the basis. When one measures a quantum state, one must measure it with respect to basis.

A loose analogy would be to think of the coordinate grid you overlay on top of some two dimensional plane you want to measure. If you want to mathematically describe a coordinate direction, it must be described in terms of a coordinate system. This coordinate system can be changed at will, and there is established mathematics to describe the way in which the math will change as a result. 

Similar to how a coordinate overlay defines what coordinate directions one can use to describe a direction, basis defines the types of measurements we can make on a system. Changing the basis one measures in will change the way a quantum state will be measured and what the outcome will be.

In fact, for every quantum superposition, there is a basis in which to measure it such that it does not behave like a superposition and there will be no probabilistic outcome.

So the question is: if we are able to affect the superposition simply by the way in which we choose to measure it, does that intentional decision change reality? It would seem to be compelling to say yes. After all, a scientist could choose to measure something in one basis rather another, and this decision has made a real change in the stuff itself.

The Quantum-Intentional Interpretation is False

So why is this so ridiculous? That's because it's a category error (you know, that same distinction that has made dualism so passé in contemporary philosophy). There are different kinds of causes, and intentions are the wrong kind of cause to truly act as the kind of explanations we want them to be.

First, a joke. One is touring a physics lab when you see an elaborate device with pieces all around the room. You are intrigued, so you ask what it does. The physicist giving your tour begins to explain each part of the device. Each step it goes through is more elaborate than the next, and it all seems to rely on mechanisms that are being studied elsewhere in the lab, but you can't figure out which part is specifically being studied. At the end of the physicists explanation, you vaguely understand the way every part functions and how they all interact with each other, but you still cannot figure out what is being studied with this device. You ask "what do you study with this device?" In reply, the physicist just laughs and says: "we don't study anything with this device, it's just an elaborate Rube Goldberg device for making coffee."

The physicist has, in a sense, answered your question. He explained what device did in terms of all of the physical workings that are involved in its functioning. The explanation that satisfied you, however, was an explanation about the intentions of those who built the device: the device is for making coffee. The joke, obviously, is that the physicist offered the physical explanation instead of the one that was most helpful. Anyone who has dealt with physicists, or similar creatures, can tell you that this not an unrealistic situation.

When we give an explanation, we can give different kinds of causal explanations. Sometimes a certain kind of causal explanation is, strictly speaking, not an adequate explanation for a question, even if it is not a, strictly speaking, incorrect causal explanation.

When scientist designs a device to measure some quantum state, of course the intentions of the scientist play a factor in how the device ends up behaving. But when a scientist is trying to understand some physical system, they are looking for a physical cause. If someone were to ask why a quantum state is showing the behavior that it is upon measurement, saying that a scientist intentionally designed the apparatus to measure in a certain basis, this would not be an adequate explanation, regardless of whether or not it is a true explanation in some sense. The physicist is looking for a physical cause, and intentions are not a physical cause.

So yes, the intentional choice of measurement basis does, in a sense, cause reality to change. However, that cause is not a physical cause, and would be an inadequate kind of explanation for a physicist studying it.

Survey of Quantum Interpretations

When I was writing my senior thesis about quantum mechanics I began to wonder which interpretations were most and least popular. My professor found this illuminating survey taken in July of 2011 at the conference named Quantum Physics and the Nature of Reality. In attendance were 33 individuals, most of whom were affiliated with physics, while a small handful were associated with math or philosophy. The study is by Maximilian Schlosshauer , Johannes Kofler, and Anton Zeilinger.

The poll is interesting to sort through. It is by no means a completely representative sample of all thinkers in the whole world, but it is sizable and professional enough to be taken seriously. It as good looking and informative graphics as well a solid statistical analysis.

Here is the link. I highly recommend checking it out yourself.

• Interestingly, they were highly divided over the nature and solution of the measurement problem. Nearly equal split of support between different answers.
• A majority of experts optimistically predicted we would have a "working and useful" quantum computer in 10 to 25 years and no one thought it would take longer than 50 years. Very few comparably think it will happen before ten years from now.
• Copenhagen is the most popular interpretation. I think that this is still an unclear result, though, since it is not always clear what the Copenhagen interpretation is. The second most popular were information-based interpretations, and I'm not entirely sure what those are. I would have guessed many-worlds would be second, but it was in a relatively close third.
• Apparently most of them thought that personal philosophical prejudice played a large part in choice of interpretation. This isn't too surprising: most of the time when one is heavily informed on a topic and debates that topic with others one is very likely to become frustrated with their views, regardless of how informed they are. While some might think this suggests that viewpoints are inherently about nothing more than our preferences, I don't think that's the right conclusion. While that could be true, I think that it suggests that smart people who strong opinions just have more opportunities to come up with excuses as to why others don't accept (their obviously true) views on some matter.
• Very few people thought we would stop having conferences devoted to quantum foundations in the future. A great many thought we would. My guess is that we will, but they will be of a completely different nature. Instead of trying to figure out the "right answer" where the focus is on debating which is best, conferences in the future will be more of a "marketplace of ideas". My reasoning: the "right answer" is not obvious to the experts now so it would definitely not be obvious to most people in the near future. Even if we discover the "right answer" in 50 years, that doesn't mean it will be commonly accepted or taken seriously. Future conferences will probably not continue to focus on sorting things out and will instead be about exploring cool ideas. I like the authors' take on this:

"Among the different interpretive camps, adherents of objective (physical) collapse theories were the only group to believe, in significant numbers, that in fifty years from now, there will likely be still conferences devoted to quantum foundations. So perhaps this reflects the fact that those who pursue collapse theories tend to view quantum theory as an essentially unsatisfactory and unfinished edifice requiring long-term modification and construction efforts. Vice versa, it may be a sign that those who regard such efforts as unnecessary or even misguided are optimistic that the remaining foundational problems, whatever they may be, will soon be resolved."

• I find it interesting that most of the correlations found in the data mostly just show logical conclusions. Not too surprising since the survey-takers are all thinkers by profession.
• The largest consensus on a question was about quantum information, where a relatively large majority agreed that it "is a breath of fresh air for quantum foundations". I'm not entirely sure what quantum information is, so I'm surprised it has been such a big deal and I haven't heard much about it.

Measurement

What exactly is "measurement"? It's a confusing question in the context of the philosophy of quantum mechanics. Confusing enough to warrant a short discussion about it.

Two Meanings

There are several different things the word could mean. The first means about the same thing as "observe". It is used to speak about the steps of an experimental setup we take to determine the state of a system or some specific property or aspect of its state. This is on par with our common-sense use of the term, like "measuring" the length of something by using a yard stick.

This first meaning also refers to when we measure something indirectly, like measuring the height of a tree by measuring its shadow and the angle the sun's rays make at the tip of the shadow. Knowing those two things you could use trigonometry to figure out the tree's height.

The second meaning is specific to the context of the philosophy of quantum mechanics. Here, "measurement" is typically meant to refer to something that happens to a quantum system in superposition. This system is put into an experimental setup that determines the state or some property of the system in such a way that it is no longer in a superposition. When a system goes from being in a superposition state to a mixed state, that is called "collapse", so this second meaning can be said to mean about the same thing as "measurement with collapse".

The Confusion Begins

What is tricky is this second meaning. It seems to indicate that there is something about observing quantum systems in superposition that will cause them to change, or "collapse". What causes this collapse, and how should we understand it?

Well the first confusion to dispel is that quantum mechanics does not show that the world around us is affected by our observations of it. This is a philosophical view (roughly, this is "idealism") which may or may not be compatible with quantum mechanics, but quantum mechanics does not in any way require this view.

One might think that it does because one conflates the two meanings of "measurement" above. In other words, one could say: 'every time we observe something, isn't some part of the universe collapsing?' The answer to this would be an unequivocal no. The world does not collapse around us because nearly nothing in our daily observable lives is in a superposition. Though material objects around us may be constituted of atoms with quantum wave functions, the superpositions associated with those atoms will have entirely unobservable effects on our lives in all but the most absurd situations.

In other words, chairs do not cease to exist in a determinate way when we leave the room as a result of quantum mechanics for a very simple reason: quantum mechanics has absolutely no say in the state of the chair. One could certainly argue that the chair does not exist when we leave the room, but quantum mechanics does not support this claim. One would have to argue this on philosophical grounds.


This discussion will be continued...

Schrödiner's Cat: Explain Live I'm Five

Recently I answered a thread on Reddit. Someone asked for the Schrödinger's cat thought experiment to be explained to them "like they were five" (basically, just use simple language and simple terms to explain something complicated). I was happy with the result, so I thought I'd share it.

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Schrödinger's cat is really strange, and that is the way it is meant to be. Schrödinger thought about this to try and make quantum mechanics seem just too weird to accept. He was wrong, though: quantum mechanics is true. So now we have a story that sounds really weird but actually reveals something hard to understand.

Matter is a Wave

The easy understand part: if you shrink down really tiny, all of matter looks like waves. These aren't sound waves or water waves, though they sometimes act similarly.

Imagine making a splash in a pool and looking at it from the side. You will see a little wave go to the left and one will go to the right. Matter looks kinda like that sometimes: a little bit of the splash goes to the left and little goes to the right. This state is called a superposition.

Now here's the weird part: even when there is only one particle, it still behaves this way. "Splash" this one particle, and it wiggles to the left, and wiggles to the right. One might ask: which way did it actually go? Well, it did not go to the left, and neither did it go to the right. It didn't literally go both directions, it just went the direction you go when you go left and right at the same time. It wiggles in both directions because it is in a superposition.

My Brain Hurts

This is weird. Lots of smart people did not like this. It made their brains hurt and they were unhappy with the idea. "How can something be in two places at once?" they would ask. Schrödinger asked this as well. He was a smart man, though, and came up with a story to help him argue.

He asked "If a particle in a superposition can move in two directions at once, then couldn't a cat in a superposition be both dead and alive at the same time?" Obviously, this made quantum mechanics sound silly.

Quantum Mechanics: Not Silly

But these people were wrong: quantum mechanics is not as silly as it sounds, it is just really really hard to understand. When a particle wiggles both left and right, it would be wrong to say that it "went to the left". It would also be wrong to say that it "went to the right". That is because it did not wiggle in only one of those directions: it wiggled in both directions because it was a wave.

Still Confused?

You might still wonder: how can something be in more than one state at once?

The answer is that it was not in several different states, it was only in one state, a superposition state. Remember, we know that the quantum wave did not go to the left AND to the right, it wiggled in both directions. It did not become two waves moving away from each other, it was still just one wave spreading apart.

You should know that no one has done this experiment with a cat. It turns out to be something that is very very hard to do. It is hard to visualize also. But the cat is made up of atoms that can wiggle in two directions. One of those directions might make the cat die, and the other direction will let the cat live. The cat is somehow in a superposition state of alive and dead. When we open the box, the cat stops wiggling and we see either an alive or a dead cat. How exactly it stops wiggling and how it becomes only one or the other is something very smart people argue about, so don't worry about that part yet.

If it seems really confusing, that is because it is.