Constructor Theory of Probability | Chiara Marletto

[00:00:00]Chiara, what is constructor theory and how does it relate to the nature of probability?

[00:00:07]>> [snorts] >> Uh so, constructor theory is a radically new way of expressing laws of physics that uses um constraints on what transformations are possible or impossible as opposed to initial conditions and dynamical laws, which is the prevailing paradigm at present.

[00:00:23]And um it's related to probabilities uh because um in fact, the structure of constructor theory is inherently deterministic in a way that it puts these constraints that are not phrased in a probabilistic way. So, when you say that that transformation is impossible, it means that um it cannot be performed by a constructor. And a constructor is an object that um can perform a transformation and stay itself, so work in a cycle after the transformation's been performed.

[00:00:55]Uh so, examples are um enzymes, computers, heat engines. They're all um instances of imperfect constructors.

[00:01:03]And when you say a task is impossible, it means that an entity of this kind cannot be built for a given transformation. Uh and when you say instead that the transformation is possible, it means that it is possible to create a constructor that can perform it.

[00:01:17]And there's no space for probability in that uh in the to the to this this sort of dichotomy between possibility and impossibility. So, how do we get probability out of the picture? Because of course, probabilities are relevant and importantly, they um are part of the structure of quantum theory or at least they are needed to describe some uh of the phenomenology that quantum systems have.

[00:01:42]And constructor theory allows you to um explain how the emergence of sto- stochasticity or the emer- the emergence of probability can um be derived from uh non-probabilistic statements. So, the the work of the constructor theory of probability is to uh define a set of conditions under which the stochasticity of quantum theory, which is expressed by the Born rule, emerges out of a set of constraints that are just about possible and impossible transformations.

[00:02:14]>> Okay, so constructor theory though seems to have this bimodal structure, possible or impossible.

[00:02:19]Um and so, at least on its first approximation, uh probability doesn't fit into that structure. I mean, cuz it's bimodal, possible or impossible.

[00:02:30]What you're saying is though, within the possible, that's how constructor theory can uh accommodate probability, which you have to do of course in quantum mechanics. Um but what what is it then add to our understanding of probability other than saying it's the possible side of constructor theory which allows it? Uh but that sounds trivial.

[00:02:56]>> [snorts] >> So, the the thing that it adds is that uh so, the first thing is that you can get um a probabilistic structure out of something that isn't probabilistic. So, that's one aspect which is non-trivial and that's very interesting in a sense.

[00:03:12]And in fact, this is uh something that's already been done for quantum theory.

[00:03:17]So, in a more specific sense, um there's been a bunch of studies that uh been made in the past where people uh have tried to derive the Born rule from axioms that are not probabilistic.

[00:03:33]>> The Born rule being that the the probability is related to the square.

[00:03:38]>> Yes, the modulus square of the wave function is the probability is related to the probability of a certain um outcome of of a certain measurement.

[00:03:45]And um and so, usually you take this as an axiom in quantum theory. So, it's just an extra assumption that you add together you know with the Schrödinger equation. It's a sort of methodological rule that tells you how to interpret repeated experiments on a quantum system in a lab. Um and if you are unhappy with the fact that you have this extra axiom which seems to jar with the fact that the Schrödinger equation is a deterministic equation, then um you might want to decompose it into deterministic axioms.

[00:04:18]And it turns out it's possible to do that. So, that's an interesting fact in itself.

[00:04:22]Um in constructor theory we do the same.

[00:04:25]It's just that it's done at this more general level, not just for quantum theory, but for a class of theories that are more general than quantum theory itself. So, that's the second non-trivial element. So, the first one is you can derive probabilities from um non-trivial uh from non-trivial non-probabilistic assumptions.

[00:04:43]And the second thing is that you can do so not just in the context of quantum theory, but you can do that for a larger class of of theories, which may include a successor quantum theory. That's why the work is relevant and interesting.

[00:04:56]>> Uh it it it sounds like what you're describing is um uh you know, pardon this expression, ad hoc. In other words, what you're describing is the way things work and everybody knows that. Now you're adding constructor theory as a as a way to uh sort of explain that, but I'm not sure I'm not sure I'm not sure and I don't see what constructor theory is adding to what we already know about how probability works.

[00:05:24]>> Um as I said, the So, the probabilities are usually um things that you use in physics as um I would say ad hoc rules to describe a variety of different phenomena. And in quantum theory specifically, you have this highly ad hoc uh axiom that appears out of the blue, which is the Born rule, which puts probabilities into the picture and, um, allows you to say that, uh, certain outcomes of certain measurements on quantum states have certain probabilities occurring given that the quantum state is a certain quantum state.

[00:06:02]Now, um, what what the constructor theory approach does is to link, um, that axiom, so the appearance of probability in a given physical situation, to axioms that are not probabilistic. So, it's a bit like saying, um, you know, you have, uh, something that is unexplained and it's just there in a theory and you explain it in terms of things that are deeper. So, that's the that's a non-trivial, um, addition to to how we think of probability.

[00:06:34]And as I said, this was done already by people like David Deutsch and David Wallace and other, uh, people who study these sort of problems within quantum theory.

[00:06:43]But the thing that was interesting to do was to do this without having an underlying dynamical structure. So, we did this in a more general way by, um, not using dynamical theories at all, but only these counterfactual statements. And it's interesting that most of the axioms that we use in order to derive, um, a Born rule-like structure are information-theoretic. So, they are not, um, subjective, they're not based on, uh, observers and other such entities, but they're really just based on these counterfactual structure that is underlying that underlies the constructor theory of information.