\"It from bit\" and the quantum probability rule by Matthew Saul Leifer

Hi Jochen,

Thanks for your comments. You are right of course that there are two kinds of noncontextuality: of probability assignments and of value assignments. Quantum theory has the former but not the latter. I think the reason why people do not talk about the former so much is that in operational approaches to quantum theory one simply defines equivalent measurements and preparations to be those that always receive the same probability, so noncontextuality of probability assignments is true by definition. This doesn't work if you are a subjective Bayesian as you can't just posit specific probabilities as a primitive concept in a physical theory.

I am aware of partial Boolean algebras and Kochen's work, as well as a whole bunch of other quantum logical derivations. It seems to me that you can either view these in a realist way as representing the true logic of the world, or as operational logics expressing connections between different experiments we can perform. I argued that Kochen-Specker presents a problem for the subjective Bayesian in the realist approach. The operational approach is more acceptable, but then I think one needs a physical explanation of why we have these structures in terms of what is going on at the ontological level.

I don't know what you mean by "only unique exists" but I'll have a look at your essay.

The lack of definitive conclusion is deliberate. All too often people take a very strong stance on the interpretation of quantum theory and see their job as defending their view against all comers. I don't think we will solve any problems this way as it leads to closed mindedness and endless debates where people are talking at cross purposes. Instead, I think the only honest thing to do is to admit that we don't have all the answers and to try and rigorously narrow down the possibilities by converting seemingly philosophical questions into concrete mathematical ones. John Bell showed us how to do this and it puzzles me that so few people follow his lead. Maybe I am more conservative and less stridently speculative than some other essay writers, but I view that as a good thing. We are trying to do science after all and the existing evidence does not lead to clean and immediate answers.

I looked at your essay. May I remind you that, on the subjective Bayesian view, probabilities only represent your degrees of belief in an uncertain event and do not require the existence of an ensemble of similar experiments. As such, they are perfectly applicable to unique events that only happen once. I am also a realist, as indicated towards the end of my essay, so I don't think we have any real disagreement on these points.

Where I do disagree is that I don't think that uniqueness is a useful scientific postulate. Science is about uncovering patterns is nature and so it necessarily has to look for commonalities rather than singular facts. I admit there may be some truths that are not discoverable by science, and uniqueness of the universe could be such a truth, but I don't see how we could base physics on that.

Hi Alexei,

1) In the Dutch book context, a decision scenario consists of a system of bets and it is a "possible" scenario if it is possible to resolve all the bets in the system together. So, for example, a bet that the position of a particle is in a certain range combined with a bet that its momentum is in a certain range is possible in classical physics, but not in quantum theory because choosing to make one measurement precludes finding out what the outcome of the other would have been. I meant to imply that this sort of thing does not only happen in quantum theory, but in any theory that obeys "it from bit" because asking one question precludes knowing what the answer to a different question might have been. I haven't read David Wallace's paper, but I'll take a look.

2) It is important to distinguish between believing that subjective Bayesianism is the best account of probability that we have and being a "Quantum Bayesian". It is slightly unfortunate that Fuchs et. al. have commandeered the latter phrase to mean a very specific brand of neo-Copenhagen interpretation because being Bayesian in itself does not imply any such thing. You can be Bayesian about the probabilities in many-worlds or in Bohmian mechanics for example. In this essay I am merely starting from the subjective Bayesian interpretation of probability and not advocating any of the other metaphysical baggage that comes with "Quantum Bayesianism". In fact, since I come down in favour of a fairly straightforward realism at the end of the essay, it should be clear that I am some distance away from Fuchs et. al.'s position. Therefore, problems that are specific to "Quantum Bayesianism", like whether Hilbert space dimension is real, are not problems for me. My problem is to figure out the best ontology, or "quantum stuff" as I called in in the essay, and this will surely render much more important things than Hilbert space dimension real. For the record though, I consider the question of ontological dimension rather silly in the foundational context, as all our fundamental theories posit only infinite dimensional Hilbert spaces. If finitism makes a comeback via some theory of quantum gravity then I will reconsider may have to reconsider, but right now it strikes me as an odd thing to fixate on. As a practical matter, some types of correlation can only exist above a certain dimension and this is what the dimension witnesses detect. However, they cannot put an upper bound on dimension, so it is still consistent to just believe that everything is in an infinite dimensional space.

Thanks. I will take a look at your essay. Unfortunately, I do not really understand the meaning of:

"I noticed that you did not mentioned how these probabilities can be distinguished as such without defining aspects of certainty"

There is probably something being lost in translation here, but what exactly do you want me to distinguish probabilities from and what do you mean by "aspects of certainty". If you can try to clarify then I will do my best to address your concerns.

Regarding "troll votes" I am inclined to believe that the opinions of the judging panel hold a lot more weight than the community ratings, so I am not too worried about it. Also, troll voters probably give everybody's essays low ratings, so they probably all cancel out in the end.

As for different senses of "it", I do think that physicists share a common language, which is the language of mathematics and empirical observation. However, when we are constructing speculative theories and explanations our views are colored by philosophical prejudice just as in any other area. Experiment will eventually determine who is right, but we need some idea of what the best directions to explore are in the meantime. It is important to note that "it from bit" is not an accepted principle of mainstream physics, but a speculative idea proposed by Wheeler. I think it is clear from his writings that it was not intended to mean "everything is discrete" or "everything is made of information", but rather "everything that appears to us to be real is a result of our interventions into nature". If people choose to interpret it a different way, either because they have not read Wheeler or because they prefer a more radical interpretation then they are entitled to do so. However, I chose to restrict attention to the principle actually proposed by Wheeler because otherwise the essay topic becomes too broad.

Regarding quantum theory as generalized probability there are a couple of points to make. Firstly, whilst it is known that quantum theory can be viewed as a generalization of probability, not everyone views this as significant, preferring to treat is as just another dynamical theory of physics. Quantum theory can be viewed in multiple different ways and this is what leads to the whole debate over interpretations. However, if we can argue, on independent grounds, that a generalization of probability is to be expected then the fact that quantum theory can be viewed in this way may take on new significance. Secondly, there is the issue of probability as a formal mathematical theory vs probability as a theory of how to reason in a world of uncertainty. It is pretty easy to write down mathematical axioms for probability, essentially it is a theory of sets of positive numbers that add up to one, but much harder to say what it has to do with the real world. Similarly, it is easy to say that quantum theory is a formal generalization of probability theory, but that does not tell us anything of foundational significance unless we can say why our reasoning about physical systems should obey that theory. My essay is really about how to fill that gap.

``It is clear that if you start with questions such as, "Is it a living object?" No."Is it here on earth?" No."Is it red?" No. "Is it round"? No, etc. You will never get the answer, 'Yes' and you must fail.''

Why?

To clarify my position, I want to make it clear that I am also a realist, as you can tell from the last section and conclusion of my essay. I am just trying to argue for this in a different style --- one that I hope is more effective against anti-realists. One can develop all sorts of a priori arguments against "it from bit" based on the idea that we should be realists. However, if you are an anti-realist then your response to this would be "What do I care? I am an anti-realist so I do not buy these arguments". To argue effectively against this, one has to start from a position that an anti-realist can support and then argue for realism on those grounds. Now, "it from bit" was posited by Wheeler as a foundational principle within a broadly ant-realist or neo-Copenhagen framework. If we can show that this principle cannot do the work required of it without being backed up with a realist conception of physics, then that should be a much more compelling argument for an ant-realist than any a priori argument for realism. That is the sort of argument I was trying to construct.

Hi Mauro,

I am flattered that you think I have philosophical training. I don't. I just read a lot of books about the philosophy of probability.

If you are a follower of Cox then it is definitely true that there is a wide gap between your position and mine. I am a subjective Bayesian in the vein of Ramsey, de Finetti, Savage, Jeffrey et. al. and I think that Cox's derivation of probability theory is one of the silliest things I have ever seen. Debating the relative merits of the two approaches could occupy a lot of space, so I will confine myself to a couple of comments.

Firstly, Cox's approach contains a lot of arbitrariness. For example, he starts from the idea that degrees of belief have to be represented by real numbers, with no real justification other than simplicity. Why do they have to be totally ordered rather than just partially ordered? Weak analogies with measuring distance with a ruler just don't cut it for me, especially since the approach does not explain how one would construct an analogous device for measuring someone's belief that would yield a real number.

Secondly, and relatedly, I believe that a viable approach to the foundations of probability has to be operational, i.e. it must say what things in the world correspond to probabilities and how to measure them. Subjective Bayesianism does this, i.e. it explains how to measure probabilities in terms of an agent's actions, but no other approach to probability really does. It is a bit complicated to explain why I think operationalism is needed here given that I am not an operationalist. Indeed, I don't actually think that probabilities ultimately should be defined in a purely operational way. It is just that, when you are confused about why a theory works, i.e. you cannot quite derive the results you need to justify the way it is applied, then it is a good idea to try to analyse the problematic concept in terms of something else and then use agreed upon facts about that other thing to see if you can find a better justification. Directly measurable things are the type of things about which we have a lot of agreed upon facts that anyone can verify, so operational definitions are the most useful for this purpose. I don't view operational definitions as "the" definition of the concept in question, but they provide a very useful rigging when there is a controversy to be resolved. As an aside, this is how I reconcile Einstein's approach to special relativity with his later statements on physics. It is not that he wanted to define spacetime operationally, but rather that he knew something had to change about the nature of space and time. The concepts of space and time come in a tight package with all the rest of the concepts of classical physics and it is very difficult to see how to unpick that package when you want to make some fundamental change. One way of getting around this is to redefine the problematic concepts, temporarily, in an operational fashion. However, after we are finished we can go back to being straightforwardly realist, e.g. viewing the structure of spacetime as the fundamental thing that accounts for the way that light rays behave rather than the other way around. It is the same with probability. We can't agree why statistics works so there must be something wrong with our usual concepts and derivations. However, probabilities are tied up with the whole theory in a tight package so it is best to temporarily define them in terms of something directly measurable. By the way, in the context of quantum theory, I think this is what Lucien means when he says that we should adopt an "operational methodology" without necessarily being operationalists.

Regarding the meaning of "context", I presume you understand that in quantum theory I intend it to be synonymous with the choice of measurement. In general, a context is the thing that determines the set of bets that can be jointly resolved. Now, of course, if we already have probability theory then we could say that there is a probability for each context and then a conditional probability for each measurement outcome given the context. Multiply the two together and you have a joint probability distribution over contexts and outcomes, which is just an ordinary classical distribution. However, the point is that we are trying to derive probability theory rather than assuming it so we have to ask what would force our beliefs about the context to be described by a classical probability distribution. I suppose you could write down an exhaustive list of all contexts and then allow bets to be made on the context as well as the measurement outcomes. Then you could apply a Dutch book to the bets on context. That would be reasonable in the 20 questions game the way I have described it in which a third party is doing the questioning. However, I also want to allow for the possibility that the bookie might be the person choosing the context and they might choose the context adversarially after you have announced your probabilities (or similarly it might be you choosing the context after making your bets and putting the bookie at a disadvantage). It might have been clearer if I had described things this way in the essay. In this case, the choice of context is not something that you can assign a probability to. Instead, you have to do a worst case analysis and hedge against all possible contexts. This type of setup is the Bayesian way of fleshing out what it means for the choice of context to be a "free choice" that we cannot assign probabilities to. Practically it just means that it might be determined adversarially so we have to do a worst case analysis.

Regarding many-worlds, I do not currently think it is a "realistic approach", but hopefully we can agree that it is a realist one (important distinction there). Although I do not advocate the theory, it remains the only interpretation of quantum theory in which a fully subjective Bayesian derivation of the Born rule along the lines I suggest has been carried out, so it would be unfair of me not to mention it. However, it is not too surprising that they are able to do this, since they start from the premise that the quantum state is real and that is the thing that carries all the information about the probabilities in the first place. It would not be too hard to derive classical probability theory if you started from the premise that reality was described by an object isomorphic to a probability distribution, and I hope we would all reject such a derivation as silly. As it happens, I am toying with a version of many worlds in which the wavefunction is not real but I still think you can derive the Born rule. I am not taking this too seriously, since it is just meant as a counterexample to the PBR theorem showing that you can have a realist theory with an epistemic quantum state if you broaden the ontology in some way. I don't think many-worlds is the best way of broadening the ontology, but one has to start somewhere and it is a more concrete suggestion than vague talk about retrocausality or "relational degrees of freedom" that you might hear from me and Rob Spekkens on other days.