# How to Rewrite the Laws of Physics in the Language of Impossibility

## Introduction

They say that in art, constraints lead to creativity. The same seems to be true of the universe. By placing limits on nature, the laws of physics squeeze out reality’s most fantastical creations. Limit light’s speed, and suddenly space can shrink, time can slow. Limit the ability to divide energy into infinitely small units, and the full weirdness of quantum mechanics blossoms. “Declaring something impossible leads to more things being possible,” writes the physicist Chiara Marletto. “Bizarre as it may seem, it is commonplace in quantum physics.”

Marletto grew up in Turin, in northern Italy, and studied physical engineering and theoretical physics before completing her doctorate at the University of Oxford, where she became interested in quantum information and theoretical biology. But her life changed when she attended a talk by David Deutsch, another Oxford physicist and a pioneer in the field of quantum computation. It was about what he claimed was a radical new theory of explanations. It was called constructor theory, and according to Deutsch it would serve as a kind of meta-theory more fundamental than even our most foundational physics — deeper than general relativity, subtler than quantum mechanics. To call it ambitious would be a massive understatement.

Marletto, then 22, was hooked. In 2011, she joined forces with Deutsch, and together they have spent the last decade transforming constructor theory into a full-fledged research program.

The goal of constructor theory is to rewrite the laws of physics in terms of general principles that take the form of counterfactuals — statements, that is, about what’s possible and what’s impossible. It is the approach that led Albert Einstein to his theories of relativity. He too started with counterfactual principles: It’s impossible to exceed the speed of light; it’s impossible to tell the difference between gravity and acceleration.

Constructor theory aims for more. It hopes to provide the principles behind a vast class of theories of physics, including the ones we don’t even have yet, like the theory of quantum gravity that would unite quantum mechanics with general relativity. Constructor theory seeks, that is, to provide the mother of all theories — a complete “Science of Can and Can’t,” the title of Marletto’s new book.

Whether constructor theory can really deliver, and how much it truly differs from physics as usual, remains to be seen. For now, *Quanta Magazine* caught up with Marletto via Zoom and by email to find out how the theory works and what it might mean for our understanding of the universe, technology, and even life itself. The interview has been condensed and edited for clarity.

### At the heart of constructor theory is the feeling that there’s something missing in our usual approach to physics.

The standard laws of physics — such as quantum theory, general relativity, even Newton’s laws — are formulated in terms of trajectories of objects and what happens to them given some initial conditions. But there are some phenomena in nature that you can’t quite capture in terms of trajectories — phenomena like the physics of life or the physics of information. To capture those, you need counterfactuals.

### Which are?

The word “counterfactual” is used in various ways, but I mean a specific thing: A counterfactual is a statement about which transformations are possible and which are impossible in a physical system. A transformation is possible when you have a “constructor” that can perform a task and then retain the capacity to perform it again. In biology, we call that a catalyst, but more generally we can call it a constructor.

In the current approach to physics, some laws already have this counterfactual structure — the conservation of energy, for example, is the statement that it is impossible to have a perpetual motion machine.

### So the constructor is the perpetual motion machine, and the counterfactual states that this transformation of usable energy to usable energy is not possible?

Yes. Counterfactuals do appear in existing laws, but these laws are regarded as second class. They are not incorporated wholeheartedly. Constructor theory puts counterfactuals at the very foundation of physics, so that the most fundamental laws can be formulated in these terms.

### How would this work in practice?

For example, consider quantum gravity. Some people say: “Why do we even need to quantize gravity given that we don’t even have experimental evidence for it? We could have a classical theory of gravity and a quantum theory of everything else.” Well, constructor theory provides us with a robust and general theoretical foundation for an experimental test that would prove that gravity must be quantum.

This test was proposed by Vlatko Vedral and me, and independently by Sougato Bose and collaborators. It goes like this: You measure the properties of two quantum masses that interact with each other through gravity only. If they develop entanglement, then you can conclude something very strong about the mediator that causes the entanglement, which is gravity. It allows you to conclude that the mediator cannot be classical — it’s got to have some quantum features.

Now, as Vlatko and I proved, the most general way to get to this conclusion is to use a constructor-theoretic principle called “interoperability,” which implies that if entanglement can be generated locally through a medium, that medium has to be quantum. It doesn’t matter in what way gravity is quantum — whether it’s loop quantum gravity or string theory or something else — but it has to be a quantum theory. It’s a test you can devise at this level of generality only by thinking in terms of constructor-theoretic principles.

### Counterfactuals seem to play a much more important or more constraining role in quantum theory than they did in classical theory. One could argue that that’s the lesson of quantum theory — that we can’t simply say “here’s what’s actually happening in the world” in some definite, unambiguous way. Is this what demands a different formulation of physical laws?

There are lots of counterfactuals in quantum theory, but that’s true in classical physics too! Quantum and classical information are two aspects of the same set of information-theoretic properties. What makes quantum information different is that it has two additional counterfactual properties.

First, it has at least two information variables — for example, position and velocity — for which it’s impossible to copy both simultaneously with arbitrarily high accuracy. Second, it must be possible to reverse any transformations on those variables.

So quantum theory has more counterfactuals, but you still need counterfactuals to fully express classical information theory and even classical thermodynamics. Concepts like work and heat can’t be captured fully with trajectories and laws of motion, because in the standard conception they are considered emergent and approximate. In constructor theory we can talk about them using exact statements about possible and impossible transformations.

### OK, so a transformation is possible if a constructor capable of enacting that transformation exists. Do you have to then give a proof that it’s possible to construct each constructor? Do you run into an infinite regress?

I think you have to turn it the other way around. Take thermodynamics. When you say that a perpetual motion machine is impossible, that statement is not something you have to prove exhaustively by checking every possible model of a perpetual motion machine, using different initial conditions and different dynamics for each one. If you had to do that, it would be a very exhausting task!

What you do is state the law in terms of possible and impossible tasks, and then work out the consequences. For instance, if you have this general statement that perpetual motion machines are impossible, you can combine it with other statements about other tasks being possible or not and work out that a heat engine is possible. And that gives you a lot of predictive power. That’s the logic. You take these statements as fundamental.

### David Deutsch defined constructor theory as the theory of which transformations can or cannot be caused and why. I’m still struggling to understand how constructor theory provides the “why.”

First, let me clarify that whether you’re using constructor theory or using the current approach, we are just dealing with some guess as to what the actual laws are. And these guesses can always be wrong. But if you buy constructor theory, the key is that just conjecturing dynamical laws will be insufficient to capture all of physical reality. You need additional principles, given by constructor theory.

So I think David is stressing that constructor theory is not just a list of things that are possible and things that are impossible. It’s the explanatory theory of why a certain pattern of possible and impossible tasks best captures what we know at the moment about physical reality. Then, if you want to question that explanatory theory, you can. But the conjecture is that whatever explanation you would come up with to improve on that, it would itself have to be expressed in terms of possible and impossible tasks.

### Would there be some primitive set of tasks at the bottom? For instance, in computation, from just a few basic logical operations, every other logical operation can be built. Do you imagine that there are a few basic constructors out of which all other constructors can be created?

In short, yes, though this is something that we haven’t really developed yet. John von Neumann, the great physicist and mathematician, conjectured a machine that was supposed to be more general than Turing’s universal computer. Von Neumann called it a universal constructor. He realized that if you think of some tasks that, for instance, living systems can do, like creating copies of themselves, a universal Turing machine cannot do that. My Mac cannot create another Mac out of some boring raw materials, even though I wish it could!

So von Neumann asked: What do I have to add to a Turing machine for it to become a more powerful machine that could construct itself? It turns out you have to add a number of things: a set of implements that allow the machine to grab the raw materials and assemble them, the ability to read instructions for assembly, et cetera.

The universal constructor is an analogue of the Turing machine in the sense that it’s supposed to be able to perform all physically allowed tasks. And we don’t know if one is possible under the laws of physics that we have. And the reason we don’t know, even 70 years after von Neumann first suggested this, is that nobody took the original proposal and connected it to physics.

Once constructor theory can define the universal constructor in physical terms and understand the principles that allow you to say that the universal constructor is possible, then we will have an answer to your question — we’ll know what are the elementary gates or elementary possible tasks that the universal constructor can appeal to when it’s trying to perform a complicated task.

### Usually we think that you start with fundamental physics and then develop technology as an application of those ideas. This almost seems to go the other way around — you start with the possibility of a technology, and that leads you to fundamental physics.

I think this is really cool. By studying the properties of something that sounds very technological — like a computer or a constructor — you end up actually studying the deepest features of the laws of physics. It’s something that fascinated me when I started studying quantum information.

Initially I thought quantum information is just some quirky application of quantum physics to computer science — but it’s not true. It’s the best tool we have to understand quantum theory itself. Measurement, EPR, entanglement: All of these things that were very puzzling even to the founding fathers of quantum theory have been worked out properly by people working in quantum information, and at the same time they were working out how to build a universal quantum computer.

It’s very cool that you can do something that’s very useful in technology, like cryptography, but at the same time you’re studying the foundations of entanglement and superpositions and so on. In constructor theory, we’re trying to follow the same kind of logic to an even more general level.

### So the technology that could come out of constructor theory would be something like an all-purpose 3D printer, capable of constructing any physical object, including a copy of itself. That’s a universal constructor.

Yes.

### Would a universal constructor also be a universal quantum computer? Or would it just have to be able to 3D-print universal quantum computers?

It’s more fruitful to think in terms of repertoires of a given programmable machine, where the repertoire is the set of tasks or transformations that the machine can perform when given the appropriate input program. In that sense, being a quantum computer or being capable of building a quantum computer given sufficient materials are essentially the same thing — because once a machine can build another machine, then the first machine has the second one’s repertoire in its own repertoire. The universal constructor has all the physically allowed computations in its own repertoire, which means it is a universal computer, too.

### And the universal constructor could even output living systems?

Yes. The physics of life would be considered a subpart of this more general theory of the universal constructor. And you could imagine how a better understanding of the constructor-theoretic foundations of the laws of physics could give you ways of programming the universal constructor to perform tasks that are relevant to that field.

So for instance, for quantum biology, you could think of the universal constructor as being programmed to mimic what happens in a plant cell when photosynthesis occurs and then thinking of ways of improving on that. You can imagine all sorts of programmable nanomachines that are specific instances of the universal constructor programmed for specific tasks. And underlying all those should be this set of principles that we will uncover by studying constructor theory. That’s the vision.

### So a universal constructor could construct a living system, but a living system itself is a kind of special-purpose constructor?

DNA is a replicator and contains the instructions for building the cell, and then the cell is the vehicle which is capable of reading those instructions, constructing a new instance of itself, copying the instructions and inserting them into the new cell. And in constructor theory you can explain why this is the only viable mechanism possible if you want reliable self-reproduction, under laws of physics that aren’t especially designed for life. So it’s not just that it’s one of the ways life can work under our laws of physics, but it’s the only way it can work. That’s a feature of living systems regardless of whether they are built with the chemistry we have on Earth.

Ultimately what we need is a theory of what makes life distinct from nonlife, and so far there isn’t a quantitative, predictive answer. What are the laws underlying this phenomenon? What’s lacking isn’t the biology — the biologists have already done their job; they’ve worked it out beautifully with evolutionary theory. Now physicists need to solve the problem within the boundaries of fundamental physics. And I’m hoping constructor theory can provide the tools to tackle that problem.