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.

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.

Quantum Bayesianism: Reaction to an Interview Part 1

In researching Quantum Bayesianism (or QBism for short) I found a fascinating interview piece online. It is a series put to the Quantum Bayesian (QBian?) Christopher A. Fuchs. While the interview is part of a book which features similar figures being asked the same questions, this interview is offered in whole for free online. I highly recommend you check it out.

Interview with a Quantum Bayesian

The following are some of my thoughts on QBism after reading the interview. ( I should note that since Fuchs is not representative for all QBians everywhere, these should just be considered impressions of QBism and not necessarily correct of all versions of QBism.) This is a continuation of my first post about QBism.

Realism and Anti-Realism

The most basic way to put QBism is that it is anti-realist about quantum states (q-states) but realist about everything else that is uncontroversially considered real in contemporary physics.

One way to explain what it means to be anti-realist about something is to say that it means one believes something is an aspect of mind and not of the world. This is not the best way, however, since delving into the philosophy of science and perception shows that this distinction is not nearly as clear as one hopes. Regardless, it is accurate to say that QBism argues that q-states are expressions of our beliefs about a quantum system and not an expression of any particular part of the system itself.

The most helpful way to think about realism and anti-realism is that it has to do with the meaning of entity-terms. Entity terms are those that we use to refer to things. Typically, an entity-term refers to something which exists. Examples of these from classical physics are "the planet Mars" or "a mass on a low-friction cart". Sometimes, however, we can use an entity term to refer to something that does not exist, however. An example of this from physics both classical and quantum is the "Del operator" (∇), which is considered only functional in use. It is a mathematical and conceptual tool and not a representation of nature (though we use it in multivariate calculus which helps us calculate things that do represent nature).

An anti-realism view of certain entity-terms would try and provide epistemic definitions for those entity-terms. This means that every situation where the offending entity-term is used can be replaced with some sufficiently long sentence describing what it means. For example, since the Del operator is used functionally, one could replace any instance of the use of it as a term with a long explanation of the mathematical role that it plays. This epistemic definition of Del operator would be considered complete because it is not considered real.

Entity-terms which we see as referring to real aspects of the world are not allowed to be constricted in this way. Though it may seem philosophically annoying, if we think of something as referring to an existing thing we must allow the term to be flexible so as to accommodate unforeseen behaviors exhibited by the real object.

Realism and Lack Thereof in QBism

All that to say, QBism considers q-states to not be real things because they can be defined as being functionally defined through the irreducible probabilities associated with them. QBism essentially argues that every q-state can be epistemically defined through the Generalized Born Interpretation. Fuchs says:

"A quantum state just is a probability assignment."

Fuchs says that he attended a conference where it was discussed what kind of linguistic classification q-states should have. In other words, are they nouns, verbs, adjectives, etc.? His answer was they are exclamations (or, more specifically, expletives). This is an absurd answer, but it should illustrate that Fuchs seeks to eliminate any talk which uses q-states.

Q-States cannot be Exclamations

The significance behind saying something is an exclamation should be immediately obvious to anyone who has taken a course in meta-ethics or has researched non-cognitivism. Non-cognitivism holds that moral sentences cannot be true or false, they are only an expression of an attitude, like "Ouch!" or "Hey!" These sentences are not considered true or false.

Sentences which use quantum states cannot possibly be exclamations because they are true or false. When a physicist uses quantum mechanics to describe a physical system he typically begins by assigning it a quantum state, and he can be mistaken or correct about the quantum state he assigns.

It should be noted that saying a q-states are a nouns does not commit one to any metaphysical view about them. Entity-terms that we do not consider actually real are still nouns. For example, one can say metaphorically that "a feeling of dread hung in the air". One can be committed to this dread not literally existing but regardless one will still use the word "dread" as a noun.

Implications of QBism's Anti-Realism

Quick question: why does science work so well? How come these weird concepts science uses can be interpreted by humans and used to manipulate the world so reliably? The easiest answer to this question: the weird concepts of science actually describe the world around us and not simply some aspect of our mind.

Anti-realist views about an aspect of science have the disadvantage of not being able to account for why those concepts are so successful at manipulating the world. Since QBism argues that q-states are not physically real things, then it must admit that it is a huge unexplained mystery as to how quantum mechanics can be used so effectively.

Surprisingly, I found this embraced by Fuchs in the interview. He says that this is the most pressing question of quantum mechanics: "Why the quantum?" (how it is possible that it works so well?). Though this is a solid argument against QBism, this shows that it is obviously no nail in the coffin. For Fuchs, it is not a problem because he sees no other interpretation of quantum mechanics as being sufficient for understanding quantum mechanics in a way that makes sense, therefore "why the quantum?" is just as much a mystery for them as well. (Whether or not he is right in this judgment is another issue entirely, but I will not deal with that here.)

For Fuchs, the ideal resolution of this is when two things are achieved:

1. All discussion of q-states is eliminated or sufficiently accounted for with epistemic definitions of meaning.
2. After reformulating quantum mechanics in this way, scientists discover what it is actually about. That is to say, it comes to light what aspects of the world quantum mechanics truly describes.

This should seem to be obviously a great deal of work to justify something simple: how an extremely useful and well-understood theory is successful in manipulating the world. Instead of simply saying "it describes the world" we must instead shrug our shoulders and then offer the above game plan to understand it. But like I said, this is no death blow to QBism, which suffers no contradiction by agreeing with this picture of quantum mechanics.

(More thoughts to come...)

Correction: In the first draft I referred to the Hilbert space as an example of an entity-term in quantum physics that is not considered "real" (by which I mean "actually existing", and not "number which could be complex but lacks an imaginary value"). Upon reflection, I think that this most likely an incorrect assumption on my part, since it seems to correspond to quantum systems and represents the possible outcomes of measurements on those systems. I'll have to look into it more to figure it out.For now, however, I replaced any mention of "Hilbert space" with mention of the "Del operator", which I am sure is functionally defined. It's kinda cheating since it is a mathematical term, but it gets the point across.