Intuition tell us that if you are able to do a measurement or experiment such that you can detect a difference, then those two items are not identical. However, this statement of identical seems to be dependant on the abilities of the experimentalist. But what about if instead of considering some classical macroscopic objects whose distinguishability is dependant on the experimentalist, we instead consider fundamental particles, such as two electrons, existing in the quantum realm. As these particles are fundamentally the same the definition of identical is no longer dependant on the abilities of the experimenter to distinguish them but is a characteristic of the particles themselves.

This quantum nature of indistinguishable has an important consequence when considering systems of these identical particles. This consequence is called the symmetrisation principle, interested readers should resort to Feynman’s lecture notes (Vol III Ch 4.). A consequence of this symmetrisation is that the state of the full system now appears to have an important quantum property, known as entanglement. This is the quantum property once described by Einstein as “Spooky action at a distance” but is now expected to play an essential part in quantum computation and communication. However, we now have a problem, despite these identical particles being formally entangled how do we utilize it when we can’t tell which particle is which?

The problem and solution is described in the following situation…

Imagine you are presented with two individuals sitting close next to each other. They are dressed completely identically wearing the same clothes, of the same size and shape and seem to always copy each other’s movements. However, you know that in reality one of the individuals is in fact not real and is just a ventriloquists dummy being controlled by its identical counterpart!

Your task is to determine which one of the individuals is the ‘real’ human being and which is just made of wood. However, when you ask the pair the question, “which one of you is real?” The skilled ventriloquist while moving his mouth and saying the words “I am” makes the dummy’s mouth moves identically. From your position it is completely impossible to determine which one of the two individuals actually said the words, and is therefore the human, and which is the dummy just mouthing along.

The identical dummies game.

What’s the solution?

Well, the problem is that with the characters standing next to each other, they cannot be addressed individually. However, if you separate the two characters such that the ventriloquist can no longer reach behind the dummy and mimic his own movements, you now have the upper hand. Most importantly you can now address the characters individually. This means that any question which you address to the dummy can no longer be faked by the ventriloquist and you can correctly determine who is who.

The same problem and solution applies to our systems of identical particles. Initially, because all the particles are all in the place, one can pretend to be the other, this is the symmetrisation principle. However, upon separating the particles the symmetrisation principle no longer applies and the particles can be individually addressed and the entanglement extracted.

In our recent work, we develop a consistent mathematical framework which fully describes the process of extracting entanglement from such systems of identical particles. A perfect example of such a system being a Bose-Einstein condensate, an incredibly cold ensemble of identical particles, know to show this interesting identical particle entanglement property. In collaboration with a group of experimentalists in Basel, Switzerland, we show that our mathematical framework consistently describes the process of extracting entanglement from a Bose-Einstein condensate and can even be used to lower bound the amount of entanglement they can experimentally extract and measure. In addition to revealing something fundamental about the nature of being identical in the quantum realm, it’s hoped that with this framework and the well-studied platform of Bose-Einstein condensates in place, the applicability and future implementation of this work speaks for itself.

The particle density of a system of particles being cooled showing the  emergence of a Bose Einstein condensate. Image credit: JPL

Earlier this year, during a morning run, I suddenly remembered about my passion for rhymes. When I was in high school and later studying Physics at the University of Salerno (Italy), I spent many a hour composing sonnets – in Italian. Some of them being love poems, others being more humorous and satirical ones to be posted on the now defunct portal Risateonline.com – often in response to challenges set by my then competitor and now friend Rullianus. Then, not sure why, but I haven’t been writing poetry in almost two decades, till now.

So the foolish idea I had during the run was to set the following challenge up to myself: for every paper published in the leading physics journal Physical Review Letters, I’d write a related poem and post it online. I am using a simple domain for this, with the acronym #PoetRL: https://poetrl.blogspot.com/

As it turns out, this year I’ve been particularly lucky with that journal, so I have had quite a lot of poetry to write in the last couple of months. If I am having fun doing so (although it takes a lot more effort for me to compose rhymes in English than in Italian), and my friends and students seem to be having fun reading the pieces, then maybe – I thought – I am not the only scientist who channels creativity by indulging in a passion for classic or modern poetry.

It then happened that a few weeks ago I saw on Twitter a call for events within the umbrella of the upcoming Nottingham Poetry Festival. Without thinking it through too much, I sent a brief description in response and came up with the idea for Science Rhymes: an evening in which a few staff and students from science-related departments at the University of Nottingham would present their own poetry inspired by the subjects they are passionate about. I didn’t really expect the idea to be too popular, as most of the colleagues I know were quick to admit they are in their comfort zone with numbers and equations, but frankly hopeless with poetic and other literary expressions. Nevertheless, a good number of people did find the idea exciting and decided to engage with it.

A big help came from the Mathematical Biology PhD student – and poet – Chris Lanyon, who was kind enough to organise a dedicated poetry workshop in the School of Maths last month, attended by a few students and colleagues, including myself. I was amazed at how expertly yet naturally he guided us to produce, on the spot, beautiful poetry, without the need for obsessive checking of Rhymezone or other dictionary tools. He taught us about some maths-inspired types of constrained writing, e.g. the “Fib”, a poetic form whose number of syllables in each row has to follow the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, …). We were also stimulated to create a more spontaneous style of poetry, combining our science specialty with particularly fond memories from our personal lives.

You will have the chance to experience all this, and much more, on Monday 29 April. I will be hosting Science Rhymes in room A17, in the Mathematical Sciences building of the University of Nottingham, starting from 5.30pm. The event is included in the official lineup of the Nottingham Poetry Festival.

The programme includes over 15 individual acts by students and colleagues, spanning several Schools in two Faculties (Science and Engineering). I have been crazy enough to prepare myself a bunch of (over 40) limericks as a way of introduction and presentation for each individual act. You can find me mostly writing poetry these days at work and at home, to be fair. Here’s one for a taste:

We’re going to explore many paths:
From Chemistry, Physics, and Maths,
     Through Neuro- and Bio-Science,
     To Computer Science.
We’ll give you some facts and some laughs.

There will be indeed a variety of  science covered, and a plethora of poetic expressions, including a song by the one and only Phil Moriarty (details on his blog), and an AI-generated poem by our own Paul Knott, who will write a dedicated post in the future to explain all about how he trained a machine learning algorithm on his writings here on Quanta Rei as well as on Shakespeare’s sonnets, in order to generate the poem.

Enticed? Attendance is free! I hope to see you next Monday at Science Rhymes.

Science rhymes with love, passion, desire,
Curiosity, brains on fire,
     Uncertainty, frustration,
     Discovery, celebration,
A drive to reach deeper and higher.

Further info:

• https://www.nottingham.ac.uk/mathematics/events/workshops/science-rhymes-nottingham-poetry-festival.aspx
• https://nottinghampoetryfestival.com/events/science-rhymes/
• https://www.facebook.com/events/423335048410979/
• https://twitter.com/GerardoAdesso/status/1103674449604292610

I recently completed a collaborative art-physics project, with award-winning artist Joseph Namara Hollis, to produce a short, illustrated book about the quantum mechanical reality in which we live.

From the start, this was an ambitious project. Our aim was to introduce quite challenging philosophical concepts deeply embedded in quantum mechanics. Our target audience was broad: we hoped that anyone from school science students to academic quantum physicists would both learn something, and be entertained, by the book. And finally, the book is short – only six double page spreads – and contains minimal text, with the intention that the illustrations play a central part in conveying the concepts we introduce. So, have we succeeded? You can find out by reading the book online here.

The seed of this project began when I undertook a short Postdoc, as part of FQXi’s physics of the observer program, at the University of Nottingham, with supervisors Prof Gerardo Adesso, Dr Tommaso Tufarelli, and Dr Marco Piani. The general question I was to dive into was: Why does our everyday world look the way it does, despite being made of bizarre quantum mechanical particles? Quantum entities, such as electrons, atoms, and photons, perpetually exist in bizarre states: they can be in two places at the same time, be travelling in multiple directions simultaneously, and can even teleport. But if everyday objects – such as chairs, tables and cats – are made of such things, then why do they look so normal?

The more specific question we hoped to answer related to the concept of objectivity. If two people look at the same object then they invariably agree on the details, such as the size, shape, and orientation, of the object. These properties are said to be objective. But, due to the bizarre quantum world discussed above, it is not clear at first sight why objects that are made of quantum mechanical particles should look objective. To cut a long story short, our work during this project, together with previous quantum physicists’ results, demonstrated that the mathematical structure of quantum mechanics itself implies that everyday objects should indeed have objective properties. As part of this project we collaborated with the same artist, Joseph Namara Hollis, to produce an illustration related to our work, which we were honoured to be selected for the front cover of Physical Review Letters.

But despite the success of this work, I wasn’t quite satisfied with the philosophical backdrop to our findings. Our results were mainly concerned with why we observe the world to be objective. But using the quantum mechanical formalism we can go beyond our mere observations, and ask questions about the whole of reality – the observed and the unobserved – in which we live. And when I plunged into such questions, I emerged with a strong view that the equations of quantum mechanics are really telling us that our universe is only one in an unimaginably large number of parallel universes! This is certainly not the only conclusion that can be drawn from the equations of quantum mechanics, but for reasons that I explain here I personally find this the most convincing and compelling.

This brings us to the main motivation of creating the FQXi-sponsored illustrated book that this blog pertains to. When we use both philosophy and physics to dive into the question of what quantum physics really tells us about reality, it turns out that there are many possible answers (of which quantum parallel-universes is just one). Other answers include that quantum mechanics must be incorrect or incomplete, and a revised theory is necessary in order to make sense of reality; or that quantum mechanics is really telling us about our knowledge of the world, rather than directly relating to the objects in our surroundings themselves. But when scouring through the literature to try to understand this, I found that the introductions and explanations to the different theories were dense and complicated, and it seemed impossibly hard to form a simple, overarching picture of what quantum mechanics tells us about reality.

This, then, is the purpose of the book: to create a (relatively) straightforward overview of the various theories that quantum physicists use to explain reality; and to introduce the philosophical conundrum at the heart of quantum mechanics that requires such an array of theories to be introduced. And to do all this is an entertaining and intuitive way by using the beautiful and fantastical illustrations of Joseph Namara Hollis.

We invite you to take a look at the book, which is freely available, and let us know what you think!

Roger Penrose’s book Shadows of the Mind confronts these problems head-on, by giving a detailed assessment of the question: Is the human mind a computer? Can it be simulated by a computer, and can its workings be understood and explained only using computation? Penrose is conclusion is a resounding no: he concludes that the human mind is not a computer, and the workings of the human mind cannot be explained by computation alone.

This book is particularly interesting to me because, having read many books and listened to many talks/podcasts on consciousness and the mind, I myself have my own strong views on the answers to these questions. My conclusion is opposite to Penrose’s, namely, it is my belief that the human mind is performing a computation. Normally a conclusion such as Penrose’s would not bother me, because many arguments can be wielded that counter such claims. But Penrose’s book is different, because at its core is a solid logical argument, one that cannot just be ignored or countered easily and quickly.

In this post I will introduce Penrose’s logical argument, and then explain how he exploits the current gaps in the unification of general relativity and quantum mechanics to justify that the human mind must utilise a new type of physics – one that cannot be simulated on a computer.

The logical argument

I will dive straight in to the key argument at the centre of this book. Consider a computation that acts on a single natural number, n. This could be something as simple as the computation “n+1”. Or it could be a vastly complicated and elaborate computation, such as the algorithm that generates the graphics of the nth level on Super Mario.

Next, we need a way of labelling all the different computations that could act on n. We say that C(q,n) is the qth computation on natural number n. It is possible to write down every single possible computation that acts on a single natural number in this way. In other words, the complete set of computations labelled C(q,n) represents every single possible computation acting on a single natural number.

The way a computation works is that it takes in an input, then it runs for some period of time, finally producing an output. When it produces the output we say it stops. However, not all algorithms stop. For example, we could have an algorithm that takes n, then adds 1, then loops around and repeats this indefinitely – i.e. it keeps adding 1 forever. We say that this algorithm does not stop. Given this, it might be of interest to know whether a given algorithm C(q,n) stops or not. To do this, we can design another algorithm, A(q,n), whose job it is to say whether the qth computation on natural number n stops:

If A(q,n) stops, then C(q,n) does not stop.

This statement is true for any values of q and n. So what about taking q to be equal to n? With this, we get:

If A(n,n) stops, then C(n,n) does not stop.                                    (1)

But notice that considering only the cases when q is equal to n, the computation A(n,n) is now only acting on a single natural number, n. Because C(q,n) includes every possible computation and a single natural number, it must also include this computation. We can label it C(k,n). Therefore,

A(n,n) = C(k,n).

Now, this is true for any value of n. In particular, it is true for the case when n equals k. Considering this value only, we have:

A(k,k) = C(k,k).                                                                                                            (2)

Now consider equation (1) above, but for the specific value k. This reads:

If A(k,k) stops, then C(k,k) does not stop.                                    (3)

Finally, combining equations (2) and (3), we have:

If C(k,k) stops, then C(k,k) does not stop.                                    (4)

What can we conclude from this? Well, C(k,k) cannot stop, because if it did then according to equation (4) C(k,k) would not stop, which would be a contradiction. But A(k,k) (which is the same as C(k,k)) is the computation that is specifically designed to determine whether C(k,k) stops. Therefore, because A(k,k) cannot stop, it is not possible for a computational procedure to determine whether C(k,k) stops. But we know that C(k,k) does not stop. Therefore, we know something that no computational procedure could ever know!

Now comes the really interesting part: we can repeat the argument above, but instead of C(q,n) representing all possible computations, we can instead think of C(q,n) as representing all possible computational procedures available to humankind. We then find:

                  If C(k,k) stops, then C(k,k) does not stop.

                  Therefore we know that C(k,k) does not stop.

But C(k,k) cannot determine this, and therefore the collection C(q,n), which represents all possible computational procedures available to humankind, cannot determine that C(k,k) does not stop. But, as said above, we know that C(k,k) does not stop. Therefore, humans cannot be using a computational procedure to determine this, otherwise there will be a contradiction.

I have skipped some of the more subtle steps in the above explanation. The more precise conclusion would say: humans cannot be using a knowably sound algorithm to determine that C(k,k) does not stop (for brevity I will not introduce what knowably or sound mean – the interested reader can check out Penrose’s book!). To very briefly mentioned some of the more subtle details, instead of concluding that humans cannot be using a computational procedure, we could instead conclude that humans are not using a sound algorithm, or a knowably sound algorithm. But Penrose very convincingly argues that we are using a sound algorithm, and it is knowable, which allows to conclude that humans cannot be using a computational procedure to ascertain mathematical truth.

But if our brains aren’t working computationally, then what are they doing? Are we using some obscure algorithm that computer scientists haven’t come up with yet, or are our minds acting as a quantum computer? In fact, neither of these options are possible, because what we mean by computation includes any process in known physics, including quantum mechanics. Therefore, the inner workings of the human mind must be beyond current physics! So what could this physics be? Is there really any room for new physics, and even if there is, can it really solve the problem raised above?

Gravity-induced collapse of the quantum wavefunction

Those readers not already familiar with the measurement problem in quantum mechanics should read my earlier blog, as otherwise this section won’t make any sense! In short, to make any sense of the process of collapse of the wavefunction – and therefore to make any sense of quantum mechanics – one must subscribe to one of the many different interpretations of quantum mechanics. To choose between them, you must decide whether you think quantum mechanics is complete/correct or not, and whether you think the quantum wavefunction represents reality, or just our state of knowledge. My own view is that quantum physics is correct, and that the theory of decoherence explains why the quantum state appears to collapse. Then, taking the uncontroversial assumption that the quantum wavefunction represents reality, I (and many others) are led to the conclusion that there are multiple parallel universes!

But different people, with different backgrounds, expertise, prejudices, and life philosophies, can take a different perspective. Physicists have not yet successfully combined general relativity and quantum mechanics into a single unified theory, so in a sense they can’t both, in their current forms, be completely correct. Given this issue, a general relativity researcher’s view could be that general relativity is correct. As with quantum mechanics, general relativity has so far never been proved wrong. But as general relativity and quantum mechanics cannot be unified, one of them, or both of them, must be incorrect and in need of modification. And given the unintuitive conclusion of assuming that quantum mechanics was correct (parallel universes!), we should assume that quantum mechanics needs modification. Many scientists might argue that general relativity is more elegant and beautiful than quantum mechanics, so many researchers would think this is a reasonable stance.

How should we modify quantum mechanics? One of the issues of combining quantum mechanics and general relativity is that it is not clear how superpositions of different gravitational fields can be understood. Roger Penrose argues that this is solved by modifying the theory so that superpositions of gravitational fields are unstable – the larger the field, the more unstable – and therefore “large” objects like humans and cats cannot exist in superposition states.

Penrose then argues that there is reason to believe that a theory in which gravity collapses the wavefunction might be non-computational. Therefore, if the human mind utilises such physics, it will be working in a non-computational way.

This line of reasoning then solves many problems in one swoop: the measurement problem is solved because gravity causes collapse of the wavefunction; one of the main problems with unifying quantum mechanics and general relativity is solved because large gravitational fields can no longer be in a superposition; and the problem of what the human mind is doing if it is not using known physics is solved because the human mind might be using the new physics that comes from gravity-induced collapse.

New physics in the brain

A popular view nowadays is that neurons are the basic “computational” element in the brain. A single neuron is a type of cell that either fires (releases an electrical signal) or does not fire, depending on the electrical signals going into the neuron from other neurons connected to it at any given time. While a single neuron is quite limited, the theory is that the 100 billion neurons in the brain, linked together by 100 trillion connections, forms a highly complicated device that produces all the fantastic thoughts and intelligent processes that go on inside our heads. This view is backed up in part by the growing evidence that artificial neural networks – which try to replicate the brain structure – can do quite amazing things, such as recognise faces or play the game of Go better than the human champion. Furthermore, it has been showed that single neurons in the brain can react to specific individual’s faces (e.g. the Jennifer Anderson neuron! https://www.nature.com/news/2005/050620/full/news050620-7.html ).

But this view of how the mind works is bad news for Penrose’s view because neurons are too large and interact too much with their surroundings to sustain any quantum coherence for a decent amount of time. In other words, a neuron cannot both fire and not fire simultaneously, in a superposition, because it interacts so readily with its surroundings that this quantum superposition is almost instantaneously destroyed (https://arxiv.org/abs/quant-ph/9907009). And in order for gravity-induced collapse to play any significant part in the brains workings, a superposition state must be sustained for a reasonable amount of time. So if neurons are the basic element in the brain that are responsible for our thoughts, then gravity-induced collapse could play no part, and the brain would be entirely computational.

However, it has certainly not been proven that the entire working of the human mind is just a neural network – to some extent this is an assumption held by current science. Indeed, neurons are not the smallest things in the brain, and it is entirely possible that each neuron is itself a miniature processor, comprised of many more basic elements that together do the work (normally a good analogy here would be that each neuron is like a miniature computer, but as the whole point is that the brain is not acting as a computer, this analogy cannot be used!).

Things certainly gets quite speculative here – and Penrose himself admits this – but the proposed site for the required non-computational processes is something called a microtubule. I won’t go into what microtubules are, because I don’t understand myself, but what I do know is that they are small enough and sufficiently ill-understood that it is possible both that they play an important part in how our brain works, and that they can sustain quantum coherence long enough so that gravity-induced collapse can play a part in how they process information. Penrose’s theory then is that gravity-induced collapse takes place in the microtubules, which means that human brain is doing something non-computational, which in turn allows us to escape the grasp of the logical argument given earlier.

Penrose’s view has many significant implications for neuroscience and artificial intelligence. Firstly, we would not be able to understand how the mind works without understanding the new physics introduced above, and as we are very far from understanding this physics, a full understanding of the mind would be a very long way off. This seems to be quite opposing to the current view in neuroscience and consciousness studies. In addition, artificial intelligence that comes even close to human cognition would not be possible without understanding this new physics. Again, this is contrary to the current feeling in the artificial intelligence community, in which most researchers are confident that artificial intelligence will continue its impressive upward trajectory and before long many human-level cognitive tasks will be surpassed by artificially intelligent agents.

Can we escape Penrose’s logical argument?

Because of the stark contrast between Penrose’s view of how the mind works, and the artificial intelligence/neuroscience views, it seems important to try to resolve the issues raised in Penrose’s book. But while Penrose’s view is relatively comprehensive, well-structured, and well thought through, it depends heavily on his logical argument. It seems that if the logical argument falls down, then there is no strong scientific reason to believe that the human mind is acting non-computationally. In turn, there is no need to introduce something radical and speculative such as gravity-induced collapse in microtubules.

For many researchers, the whole theory seems to rest somewhat precariously on this logical argument, and it is highly tempting to just disregard it or ignore it. Unless I’m mistaken, this is potentially what almost the entire artificial intelligence and neuroscience communities are doing! To some extent this is understandable, because there aren’t many people in the world who understand logic and computation enough to thoroughly study Penrose’s argument in order to try and find a flaw in it. Until recently, I too was culpable of strongly believing that the human mind was a computer, without myself trying to find a flaw (or trying to find others who’d found a flaw!) in the logical argument. For such a large chunk of scientists to ignore his argument might turn out to be a risky strategy, and arguably more researchers should invest in studying his points more carefully.

I should mention here that after Penrose’s first book on this topic, The Emperor’s New Mind, in which he introduced a similar but slightly weaker logical argument, many researchers tried to find flaws, but Penrose has seemingly quite successfully batted these all away in Shadows of the Mind with a comprehensive half-chapter in which he carefully and thoroughly addresses 20 of such arguments. It’s probably safe to assume that Penrose also received many attempts at counter-arguments after his second book, but I know that Penrose himself has been unconvinced by these, and there is certainly no widely agreed upon and well understood counterargument that the rest of the scientific community wield in order to reject Penrose’s views.

Having now read the key parts of Shadows of the Mind twice, and spending some time pondering it, I myself now have many ideas for where I think the logical argument could fall down. However, I am not a logician and I have certainly not spent enough time to confidently say that I thoroughly understanding the argument, so while there is a remote chance that my ideas are correct, it’s probably more likely that they are not! So I’ll refrain from presenting them here, but I’m very happy to discuss them with anyone interested.

I will, however, present the core of a counter-argument introduced in a very interesting review of Shadows of the Mind by one of the most important consciousness researchers, David Chalmers (http://journalpsyche.org/files/0xaa25.pdf) . The general structure of Penrose’s argument involves taking a number of assumptions, then through the steps presented above he derives a contradiction. In Shadows of the Mind he quite convincingly argues that of all the assumptions are reasonable, except the assumption that the human mind acts computationally. Having derived a contradiction, at least one of the assumptions must be dropped, so Penrose drops the computational one. But Chalmers presents an argument (by McCullough and Löb) that takes all of Penrose’s assumptions, without the one about computation, then after a number of steps a contradiction is derived. Therefore, one of these assumptions is wrong. But this means that one of Penrose’s assumptions – and crucially not the one about computation – is incorrect, and Penrose should drop this one rather than his one about competition. Then there is no reason – at least not based on this logical argument – to think that the human mind is not acting as a computation.

Conclusion

I want to finish on a high, so I will end by saying that this is an exciting and fascinating book to read. It is clear, well-written, entertaining, and thorough. Given the conclusions alone, my first reaction was that they are implausible and outlandish. But following his arguments step-by-step, he only rarely takes radical or outlandish steps. If he is wrong, then he is only subtly wrong, so naturally the reader is led along a path that eventually leads to his conclusions. I still disagree with them, but I had an enlightening and thoroughly enjoyable time reading his book, and trying to work out which subtle points in the logical argument I disagree with, so that my own view – that the human mind is a computer – can be kept intact!

Meta-theoretical knowledge

If you are familiar enough with quantum information, I bet you have already heard of at least one meta-theoretical statement — a very important one. Before we come to that, let us give some examples to clarify the above tripartite categorisation of knowledge. Since this blog is about quantum mechanics, that will be our arena. In this context, our knowledge of the double-slit experiment and of its profoundly surprising results is undoubtedly empirical. Building on this and other experiments, physicists have come up with a mathematical theory of physical phenomena at the microscopic level, called quantum mechanics. Quantum mechanics, with its internal structure made of Hilbert spaces, wave functions, Born rules for measurement and collapse, and so on and so forth, is among the most luminous and successful physical theories, and thus belongs to the second layer of knowledge. One of the most striking phenomena whose existence is predicted by quantum mechanics is entanglement, i.e. the fact that distant particles may be left by previous interactions in a state that produces impossibly strong correlations upon local measurements on the two particles. The precise meaning of the words ‘impossibly strong’ was clarified by J.S. Bell in his groundbreaking 1964 paper: those correlations cannot be explained by any local hidden variable model, i.e. they are fundamentally non-local. I want to argue that this deduction — nowadays known as Bell’s theorem — is the prime example of a meta-theoretical statement. To understand why, note that Bell’s theorem has basically nothing to do with quantum mechanics; rather, it concerns its experimental predictions. It is these predictions, and not merely quantum mechanics, that are claimed to be incompatible with a local hidden variable model. In other words, any alternative theory that happened to predict the same experimental outcomes with the same probabilities (at least in the context of a Bell test) would run into the same problem, the existence of non-local correlations. Since Bell’s theorem applies to a whole class of theories, i.e. those whose experimental predictions of the outcomes of a Bell test coincide with those of quantum theory, it is a truly meta-theoretical statement. Not only: its importance lies precisely in this feature. Every person who is familiar with the formalism of quantum mechanics can see that it is intrinsically ‘non-local’, because of the way post-measurement collapses work. But this observation alone is not worth much; what if the true theory of Nature were not quantum mechanics? We cannot draw conclusions about the deepest mysteries of Nature based on our formalism; instead, we need meta-theoretical statements to certify that the bizarre phenomenon we are facing does not admit a more ‘mundane’ explanation in the context of some alternative theory.

What is a physical theory?

This discussion should convince the reader of the conceptual importance of meta-theoretical results. That being said, how do we study a set of theories that we do not even know? How do we formalise the abstract concept of a ‘physical theory’? An answer to these questions is provided by the machinery of general probabilistic theories, which we now set out to describe.

In quantum information and more generally in information theory it is common to take an operational approach to problems: a concept is defined and characterised by what allows you to do. Thus, a physical theory is for us simply a set of rules that allow to deduce a probabilistic prediction of the outcome of an experiment given the detailed description of its preparation. The above definition is quite broad. Let us discuss some of its main features. For a start, (i) ‘probabilistic’, as opposed to ‘deterministic’, refers to the fact that it is conceivable that the theory determines only the probabilities of recording the various outcomes rather than producing a definite prediction of what the outcome will be. This is the case, for instance, for quantum mechanics, which we definitely want to consider as a legitimate theory. A second observation is that (ii) the above definition depends on the set of experiments under examination; the smaller the set, the larger the number of possible theories. This is in general a desirable feature, as long as we want to be able to declare e.g. Newtonian gravity a fully-fledged physical theory, although it can only explain the motion of slow massive bodies and gets into trouble when one brings electromagnetic phenomena or relativistic speeds into the picture. A third remarkable feature of our definition of a physical theory is that (iii) it does not necessarily describe systems that evolve in time. Instead, it takes an ‘input-output’ point of view: an experiment is described as a procedure starting with a preparation of a physical system and ending with the reading of some digits on a screen. Of course, we can decide to track time evolution by designing a whole set of experiments on identically prepared systems ending at different times, but this need not be the case; depending on the context, we can content ourselves with a much more essential description of a more limited set of experiments.

Ludwig’s theorem

With this premise, we can now study the simplest physical theories of all: those that describe a single system. Such a system — that can be an atom, an electron, but also Schrödinger’s cat, depending on your stance on the issue of animal rights — can be prepared in a variety of ways, labelled by an index ω ∈ Ω, where Ω is the set of all possible preparations (hereafter called ‘states’). Also the measuring apparatus that we employ to measure the system can be set up in different ways; each 𝝀 ∈ Λ will describe a specific outcome corresponding to a particular setting of the apparatus (hereafter called ‘effect’). In this context, a theory is simply a function 𝜇 that takes as inputs a preparation ω and a measurement outcome (together with the description of the setting of the measuring apparatus) 𝜆 and outputs a probability, i.e. a real number between 0 and 1. In mathematical terms, we write 𝜇 : Ω × Λ → [0, 1].

It may be useful to pause for a second now, and describe the familiar example of quantum mechanics in this language. For the sake of simplicity, we consider a single qubit, i.e. a 2-level quantum system. The set of states Ω comprises all 2×2 density matrices and is well-known to be representable by a 3-dimensional ball — called the Bloch ball. The shape of the set of effects Λ depends on the actual measurements that are available in our laboratory. In the best-case scenario, we can assume that all so-called positive operator-valued measurements (POVM) are experimentally accessible. When this happens, effects are 2×2 matrices 𝝀 that satisfy 0 ≤ 𝝀 ≤ 1, where 1 stands for the identity matrix, and the inequalities are to be intended in the sense of positive semidefiniteness. Finally, the function 𝜇 encodes the Born rule: 𝜇(ω, 𝝀) ≔ Tr[ω 𝝀].

In the most general (non-quantum) case, the function 𝜇 as well as the sets Ω and Λ are a priori unconstrained. However, if they are to describe a physical system, it is very reasonable to postulate that they satisfy some further properties, purely on logical grounds. We formalise these properties as axioms below.

Axiom 1. The function 𝜇 separates points of Ω and Λ, namely, if some ω₁, ω₂ ∈ Ω satisfy 𝜇(ω₁, 𝜆) ≡ 𝜇(ω₂, 𝜆) for all 𝝀 ∈ Λ, then it must be ω₁ = ω₂ . Analogously, if for given 𝝀₁, 𝝀₂ ∈ Λ one has that 𝜇(ω, 𝝀₁) ≡ 𝜇(ω, 𝝀₂) for all ω ∈ Ω, then 𝝀₁ = 𝝀₂ .

In other words: if two states can not be distinguished by any effect, then to all intents and purposes they are the same state. Analogously for the effects. This assumption is not really fundamental. On the contrary, given any pair of sets Ω and Λ and a function 𝜇 that does not satisfy Axiom 1, we can always define two modified sets Ω’ and Λ’ by identifying the elements that are not separated by 𝜇. The resulting quotient function 𝜇’ then satisfies Axiom 1  on Ω’ and Λ’ by construction.

Axiom 2. There is a ‘trivial effect’ u such that 𝜇(ω, u) ≡ 1 for all ω ∈ Ω. Moreover, for all 𝜆 ∈ Λ there is an ‘opposite effect’ 𝜆’ such that 𝜇(ω, 𝜆’) ≡ 1 − 𝜇(ω, 𝜆) for all ω ∈ Ω .

This axiom is also very natural. The trivial effect can simply be obtained by preparing the measurement apparatus to produce a fixed output. Given an effect 𝜆, setting up the apparatus exactly as described by 𝜆 and accepting all outcomes that do not correspond to 𝜆 yields its logical negation 𝜆’.

Axiom 3. Probabilistic mixtures of states (respectively, effects) are again valid states (respectively, effects). That is, for all states ω₁, ω₂ ∈ Ω and probabilities p ∈ [0,1], there exists a state 𝜏 such that 𝜇(𝜏, 𝜆) = p 𝜇(ω₁, 𝜆) + (1 – p) 𝜇(ω₂, 𝜆) for all 𝜆 ∈ Λ. Same for the effects.

What Axiom 3 is telling us is that given two preparation procedures ω₁, ω₂, it should be possible to construct their probabilistic combination. This is a third preparation procedure     𝜏 obtained by flipping a (biased or unbiased) coin, preparing the state according to ω₁ or ω₂ depending on the outcome, and then forgetting that same outcome. Clearly, 𝜏 must reproduce probabilistically the behaviour of ω₁ and ω₂ when tested on all effects 𝜆.

The formalism we have developed so far seems to be rather inconvenient, especially when compared with the quantum mechanical example described above. Let us identify its main weaknesses. (a) In quantum theory, states and effects are represented by objects in a linear vector space; this means e.g. that the state 𝜏 of Axiom 3 is simply given by the convex combination p ω₁ + (1 – p) ω₂ , which eliminates the need to go through the function 𝜇 to determine it. (b) The quantum Born rule 𝜇(ω, 𝝀) = Tr[ω 𝜆] is bilinear in the state and the effect (meaning that it is linear in each argument when the other is kept fixed). This is a very convenient property especially when long computations are involved.

To fix these inconvenient features of the 𝜇-formalism, a theorem by the German physicist Günther Ludwig comes to our rescue. It states that whenever Axioms 1,2,3 are satisfied, the sets Ω and Λ can be thought of as convex sets in linear vector spaces, so that (a) probabilistic mixtures become convex combinations; and (b) the 𝜇 function (generalised Born rule) is bilinear in the state and the effect. For completeness, we state the theorem formally below.

Theorem (Ludwig’s embedding theorem). If Axioms 1,2,3 are satisfied, there is a vector space V such that:

• Ω ⊆ V, Λ ⊆ V^* are convex sets, with 0,u ∈ Λ, and Λ = u − Λ;
• ℝ₊∙Ω = C is a cone, ℝ₊∙Λ = C^* ≔ { 𝜑 ∈ V^*:  𝜑(x) ≥ 0  ∀ x ∈ C } ⊆ V^* is the corresponding dual cone; the set of states satisfies Ω = {x ∈ C : u(x) = 1};
• the generalised Born rule holds: 𝜇(ω, 𝝀) = 𝝀(ω).

Here, the symbol V^* represents the dual of the vector space V, i.e. the space of all linear functionals acting on V. Hereafter, we shall refer to C as the cone of unnormalised states, and to the functional u as the order unit. The dual cone C^* lives in the dual space, and can be simply thought of as formed by all linear functionals that are positive on the whole C.

Note. For the sake of readability, we stated the theorem in the simple case where V is finite-dimensional. It should be noted, however, that one of the big conceptual contributions by Ludwig was to formulate and prove an embedding theorem that covers also the infinite-dimensional case, and is (unsurprisingly) much more technically involved.

Ludwig’s embedding theorem provides an elegant solution to the main problems of the 𝜇-formalism as discussed above. It is so important because it provides the foundation for the formalism of general probabilistic theories (GPTs). The theorem was proved long ago, in a series of works published between the 1960s and 1970s. Nowadays, it is implicitly used in practically any paper on GPTs. It is indeed common to start any analysis by simply assuming that the state space is a convex set in a real vector space, and that the effects are linear functionals with values between 0 and 1 on said state space. That this assumption causes no loss of generality is precisely the take-home message of Ludwig’s theorem.

The formalism of general probabilistic theories

We conclude this post by summarising the formalism of GPTs as it emerges from Ludwig’s theorem (and adding the last bits and pieces). It is useful to start with a picture.

1. The cone formed by unnormalised states (positive multiples of physical states) is a proper cone C in a real vector space V . Here, ‘proper’ means that it is closed, convex, that it spans the whole space (we want to exclude the ‘flat’ case), and that it does not contain any straight line (opening angle less than 𝜋).

2. The order unit u belongs to the interior of the dual cone C^*, in turn defined as C^* = {𝜑 ∈ V^*: 𝜑(x) ≥ 0 ∀x ∈ C} ⊆ V^*;

3. The state space is obtained by slicing C with u at height 1, in formulae    Ω = {x∈ C : u(x) = 1}. Statistical mixtures of states are represented mathematically by convex combinations.

4. Measurements are represented by (finite) collections of effects ( e_i )_i∈I ⊆ Λ that add up to the order unit, i.e. such that Σ_i∈I e_i = u. The probability of obtaining the outcome i when measuring the state ω is given by the generalised Born rule: p(i|ω) = e_i (ω).

In stating (4) we made the implicit assumption that all linear functionals that take on values between 0 and 1 on Ω are legitimate effects. This assumption is commonly known as the ‘no-restriction hypothesis’, and corresponds geometrically to requiring that the effects Λ cover the whole red set on the right side of the above figure. This is not necessarily the case in many conceivable situations. For example, it may happen that our experimental apparatus is subjected to some intrinsic noise that makes sharp measurements impossible. The main justification of the no-restriction hypothesis, beyond its mathematical simplicity and elegance, lies in the fact that it is well-verified at a fundamental level in classical and quantum theory. That is, we believe it possible, at least in principle, to construct a close-to-ideal measurement apparatus that can implement any given sharp quantum measurement. Unless otherwise specified, we will always make the no-restriction hypothesis from now on.

We conclude this post by discussing some examples of GPTs.

• Classical probability theory. The system is simply a ball hidden somewhere in one out of d boxes. States represent our partial knowledge of the ball’s position, and are thus modelled by probability distributions over a discrete set of size d. The host vector space V = ℝ^d comprises all real vectors of length d, and unnormalised states form the cone ℝ₊^d of vectors with non-negative components. The order unit is the functional acting as u(x) = x₁ +…+ x_d , so that the slice u = 1 of the cone ℝ₊^d truly identifies the state space Ω as the set of all probability distributions. Its geometrical shape is that of a simplex, more precisely, a regular (d-1)-simplex. To picture a simplex in your mind, note that a regular 1-simplex is a segment, a regular 2-simplex an equilateral triangle, a regular 3-simplex a Platonic tetrahedron, and so on.

• Quantum mechanics. We discussed this important example before in the simple case of a qubit. In the quantum mechanical theory of an n-level system, the host vector space comprises all n × n Hermitian matrices. Although some of their entries may be complex, Hermitian matrices form a real vector space, closed under real (not under complex!) linear combinations. The cone C of unnormalised states is formed by all positive semidefinite matrices, and it is sliced at height 1 by the order unit functional u(X) = Tr[X] (the trace). The resulting section is the set of all n × n density matrices, i.e. positive semidefinite matrices of unit trace; we thus get back the familiar quantum formalism.

• Spherical model. As already mentioned, the state space of a single qubit is geometrically representable as a ball — the Bloch ball. This is really a peculiar feature of dimension 2, the set of quantum density matrices in higher dimension possessing a more complicated structure. However, we can take inspiration from this considerations to design a GPT whose state space is defined to be a d-dimensional ball. The way to obtain this in the GPT formalism is very easy, and is already pictorially represented in the above figure. We can take as host vector space V = ℝ^d+1 (all real vectors of length d+1), and as C the ‘ice cream cone’ formed by all vectors x = (x₀, …, x_d) such that x₀ ≥ (x₁² +…+ x_d² )^1/2 . Then, the functional u(x) = x₀ does the job of slicing C so as to produce a d-dimensional Euclidean ball.

• Generalised bit. Now that we have in our toolkit a GPT whose state space is a Euclidean ball, why not construct one whose state space has a cubical shape? More precisely, we want Ω to be the d-dimensional hypercube, which can be mathematically defined as the convex hull of all points in ℝ^d with coordinates ±1. This is more easily done that said. Let the ambient vector space be V = ℝ^d+1 (as before), and take C as the cone formed by all vectors x = (x₀, …, x_d) such that x₀ ≥ max{x₁ ,…, x_d }. It is not difficult to verify that the same order unit functional u(x) = x₀ determines on C a section shaped as a d-dimensional hypercube.

This concludes our introduction to the modern formalism of general probabilistic theories. We have seen how this emerges naturally from a very broad definition of the notion of physical theory via Ludwig’s theorem, and we discussed a few (hopefully instructive) examples. For those who would like to know more about the early days of the GPT formalism, and especially delve into some of the mathematical details, I warmly recommend having a look at [1]. The original proof of Ludwig’s theorem can be found in [2] and references therein. I tried to give a self-contained account of it with all the technical details in Chapter 1 of my PhD thesis [3]. Chapter 2 of the same thesis summarises instead the GPT formalism in finite dimension.

[1] A. Hartkämper and H. Neumann. Foundations of Quantum Mechanics and Ordered Linear Spaces: Advanced Study Institute held in Marburg 1973. Springer Berlin Heidelberg, 1974.

[2] G. Ludwig. An Axiomatic Basis for Quantum Mechanics: Derivation of Hilbert space structure, volume 1. Springer-Verlag, 1985.

[3] L. Lami. Non-classical correlations in quantum mechanics and beyond. PhD thesis, Universitat Autònoma de Barcelona, October 2017. Preprint arXiv:1803.02902.

Yet for those doing science with a passion, beautiful equations are perceived as equally if not more touching than a work of art, and creativity is a fundamental requirement for reaching beyond the state of the art on our understanding of how the universe works. Similarly, an increasing number of artists resort to analytical methods to explore novel ways to communicate inner expressions of self. And let’s not forget that a streak of polymaths, in which a combination of scientific and artistic skills has resulted in universally acclaimed masterpieces which made history, can be traced back many centuries: Leonardo da Vinci is perhaps the brightest example.

In contemporary times, despite a somehow worrisome cultural crisis, I like to think that the community of laypeople who do not actively engage in science or arts as their main profession, but who nevertheless display an interest and a capacity to appreciate advances in both domains — and at their interface — is in fact on the rise.

Quantum science, which is my specialty, seems to be one of the subjects which is especially likely to capture the public’s attention. I was once told that, when popular magazines such as New Scientist or Scientific American put the word “Quantum” on their cover, often accompanied by a catchy drawing, it is almost guaranteed that sales of that particular issue will be bigger than the average. While this has led to misuse and sometimes abuse of quantum concepts in such diverse areas ranging from ‘healing’ to household cleaning, it has also stimulated scientists and artists to find more creative ways to merge the best of both their worlds to engage in better dissemination.

In a previous post, I spoke about “Entanglement”, the painting which was gifted to me by artist Pam Ott and which has inspired my own research on quantum entanglement in the past decade or more.

More recently, we had another successful cross-fertilisation between maths and visual arts, with a drawing designed by local artist Joseph Hollis (see more of his science-related work here) to accompany our research on Quantum Darwinism, which was selected by the American Physical Society to be featured on the cover of the October 19 issue of the prestigious journal Physical Review Letters – a great honour for us! The University of Nottingham Press Office covered the news with a blog post entitled “Maths inspired art makes front cover”, available to read here. Curiously, when tweeting about it, the Press Office fell perhaps into a Freudian slip and wrote “Art inspired maths” instead…

While probably unintentional, this hints at how deeply science and arts can mutually influence each other and continue to progress to uncover new horizons.

Our group’s collaboration with Joseph Hollis took to a new level with the very recent production of an illustrated book on the foundations of quantum mechanics. Written by Paul Knott and funded by the Foundational Questions Institute, the book “Our quantum reality” is a gorgeous example of how efficiently (cat-themed) figurative arts can assist science and scientists in reaching out and delivering even complex and mindboggling messages. You can read more and flip through the entire book at the dedicated webpage: https://illustratedquantum.wordpress.com/

What’s next then?

Science and arts will meet in an even grander love story later this month. On Tuesday November 27^th, at the Djanogly Theatre, Lakeside Arts Centre, set in our beautiful University Park Campus (University of Nottingham), we will have the pleasure to host a performance of “Entanglement: An entropic tale!”. A fully-fledged opera produced by Roxanne Korda (librettist) and Daniel Blanco Albert (composer) from Birmingham-based company Infinite Opera, inspired by concepts of modern (quantum) physics!

This is the first time I have ever heard of entanglement and entropy mentioned in a theatrical piece, and I am really excited about the upcoming show — described as the “Romeo and Juliet” of particle physics — and the emotional journey it will take its viewers on! Here is me chatting with Roxanne and Dani at Lakeside Theatre a few weeks ago, in preparation for the event…

I will be introducing the opera with a brief presentation, in which I hope to convey successful instances of breaking the (virtual) boundaries between science and arts, as I am arguing in this post. For those of you coming on the evening, there will also be the chance to ask more questions on this and related topics during a Q&A panel session after the performance. For this we will be joined by Professor Mark Fromhold from the School of Physics and Astronomy (University of Nottingham) who has done his fair bit of taking down the science vs arts apparent wall by collaborating with songwriters Lady Maisery on their song Order and Chaos (which mentions entanglement!), by engaging with public outreach at Pint of Science events and more recently at the Royal Society Exhibition on “Quantum sensing the brain”, as you can see here.

If you are in the area and have not booked your tickets yet, please do so! We look forward to welcoming you to an evening which science and arts lovers will hopefully both find intriguing and delighting.

P.S. You might also have the chance to buy a copy of “Our quantum reality” for a very small charge on the occasion of the opera night.

Tickets available here: https://www.lakesidearts.org.uk/music/event/3907/entanglement-an-entropic-tale.html

Quantum Roundabout 2018 was the fourth iteration of a series of dedicated conferences looking at foundational mathematical topics within quantum physics.

The etymology of the conference stems from the so called “Magic Roundabout” on the outskirts of Swindon. This roundabout boasts five mini-roundabouts arranged around a sixth central anticlockwise roundabout.  It was decided that such a strange junction could only be quantum in nature, hence the conception of Quantum Roundabout. The junction has since been named fourth scariest in Britain by a recent poll .

Held at the University of Nottingham every two years, Quantum Roundabout brings together early-career researchers from around the world to collaborate and engage in scientific discussion. The organisation of this conference is a rite of passage for all PhD students under the supervision of Professor Gerardo Adesso, who has overseen the organisation of all four iterations of the conference. Since its conception, attendance to Quantum Roundabout has grown significantly, with the most recent edition boasting over seventy attendees.

A specific motivation behind Quantum Roundabout is to provide a platform for early-career researchers to communicate their research and engage in scientific discussion, with a focus on new and emerging topics within quantum theory.
Each of the three days of the conference focused on a different theme with a morning and afternoon tutorial lead by an expert in the field. This year’s Quantum Roundabout covered the following areas of interest:

• Foundations of quantum thermodynamics. Prof. Jonathan Oppenheim (UCL) delivered tutorials exploring the ultimate limitations for thermodynamic physical processes as formalised within the mathematical formalism of quantum resource theories.
• Mathematical frontiers in quantum information. Nilanjana Datta (University of Cambridge) drew our attention to the structure of Shannon theory and her recent contributions to majorization theory.
• Resources for quantum technologies. Prof. Matteo Paris (University of Milan) focused on the mathematical and statistical tools underpinning the implementation of quantum estimation theory and its range of applications.

Each of the tutorials underpinned the theme of the day, around which participants gave talks based on their current research.

A key goal of the Quantum Roundabout conference is to highlight and promote the role of mathematics in the approach to quantum theory in both its fundamental and applied aspects. 

Indeed, mathematics is not just a tool when tackling a physical problem; sometimes it is the motivation behind the research, other times a mathematical theorem could be the ultimate result and, most of the time, the mathematical rigour becomes crucial in the understanding of physical concepts. This has been true in the past and still is today.

Recently, for example, the mathematical structure of resource theories, which has its roots in convex geometry, has found application in the burgeoning field of quantum thermodynamics. The classification of equilibrium states as free states and the energy preserving operations as free operations gives the laws of thermodynamics a mathematical foundation.

Information theory, originated by Shannon’s pioneering work, has provided versatile foundations for a variety of fields, which have attracted an increasing interest in recent years thanks to their potential for technological breakthroughs.

One of the most promising such technologies is quantum metrology, which studies the exploitation of quantum mechanical effects to develop measurement techniques that are higher in sensitivity or resolution than possible using classical systems alone.

As expressed by participants and the invited speakers, Quantum Roundabout 2018 delivered an effective platform for dedicated scientific discussion on emerging topics within the field. This atmosphere of discussion has since led to several collaborations between early-career researchers at international institutions. It is hoped that the continued success of Quantum Roundabout will help fuel future directions in the field of quantum theory and cement the University of Nottingham as a centre for quantum research.

This conference would not have been possible without the financial help of the IMA in addition to our other sponsors (LMS, JPhys.A, IAMP, Uni of Nottingham, IOP, Xanadu and FQXI). The organisers also wish to thank Prof. Gerardo Adesso, Dr Ludovico Lami, Dr Paul Knott and Katie Gill for their advice and support. Benjamin Morris would also like to thank his fellow organisers, Giorgio Nocerino, Buqing Xu and Carmine Napoli for making this year’s Quantum Roundabout a success.

Written by Benjamin Morris

To be more specific, our task is the following:

Given a set of quantum-optics experimental equipment, what is the best way of arranging the apparatus to create a quantum state with certain desirable properties?

What we mean by “desirable” depends on the application in mind. We are interested in designing quantum states of light, which can be useful range of applications such as quantum computing, making high precision measurements, and quantum cryptography. Now, the apparatus we are using can be broken into three categories: states, operations, and measurements. Roughly speaking, we take some states, act on these states with some operations, which modify the states and cause them to interact with one another, then we perform some measurement. This is getting rather complicated, so to simplify things we will use Lego!

The meanings of all the symbols in the white boxes is not so important for the purpose of this blog, but for those already familiar with quantum optics we provide a glossary at the end of this blog that says what all these different symbols means. If we now think of this in terms of Lego, our question becomes:

Given a collection of Lego pieces, what is the best way to arrange them to produce a construction with certain properties?

Just like in quantum optics, there can be a variety of desired Lego constructions, such as making something strong, something beautiful, or making something that looks like a real-life object such as a police car. And just like with quantum experiments, the usual way to design a Lego construction is to use creativity, prior knowledge, and intuition.

But how could a computer design a new Lego construction? The technique we use is known as a genetic algorithm. Genetic algorithms are designed to mimic natural selection, and they work as follows:

First, the computer makes a collection of completely random constructions. This collection is known as the initial population:

We then assess the various designs, giving them all a score:

Next, we select the best designs, and throw away the ones that aren’t so good:

This leaves us with a smaller subset of the population, which we call the parents:

We will use these parents to create a new population, known as the children. First, we take the very best parents, and copy them without any modification. This produces the elite children:

Second, we breed some of the parents together to produce crossover children:

Thirdly, we make small changes to some of the parents, producing the mutation children:

This leaves us with a new population, which hopefully should be significantly better than the initial population:

We then repeat this process:

Each cycle is known as a generation, and after repeated generations we create better and better individuals. Just like in nature, eventually we end up with individuals that are highly suited for our goals (try and work out what the goal was here!):

And the result is the same for us: after a number of generations, our genetic algorithm produces a range of new quantum experiments that often outperform the human-designed experiments!

Acknowledgement: It should be stressed that this post is based on a poster by Rosanna Nichols, who should take most of the credit! A paper based on the genetic algorithm introduced here will be available shortly, but the earlier paper can be found here.

Glossary:

Superintelligent machines

A superintelligence can be defined as an “agent that possesses intelligence far surpassing that of the brightest and most gifted human minds” (Wiki). Intelligence itself is hard to define, but for our purposes it is fine to use a broad definition that encompasses every possible type of intelligence that you can think of: IQ tests, problem-solving, artistic ability, creativity, game playing, emotional intelligence, and so on. It is already clear that computers can surpass humans in certain, very specific types of intelligence, such as playing chess. There are also a few general things for which computers are vastly superior to humans. Computers are much faster at repeating simple tasks, such as calculations, as can be demonstrated by picking up your calculator and crunching some numbers. I have also found this aspect very useful in my own research, where I used a simple AI to design quantum physics experiments. Furthermore, computers can store and process vast amounts of data – imagine searching on the internet without Google! Note that we normally don’t refer to Google search as “intelligent”, but imagine the reaction when telling the average 19^th-century scientist of its abilities.

Despite these successes, the majority of tasks performed by humans are still far superior than those of AI. Simply making a cup of tea, especially in an unfamiliar kitchen with unfamiliar equipment, is challenging for an AI, but trivial for us. The difference often comes down to common sense, of which we have plenty, but AI lacks entirely. For this reason, and others, it is at present hard to imagine an AI that could function in the world as well as we can. But is it impossible for us to build a human level, never mind a superintelligent, AI?

In the short/medium term, there seems to be no reason why development in AI won’t continue its rapid upwards trajectory. AI has already come close to (and sometimes outperformed) humans in tasks that reasonably recently seemed impossible, such as playing Go, competing in quiz games such as jeopardy, driving cars, and language translation. And with AI now proving its genuine commercial potential, it is reasonable to assume that funding in AI will continue for the foreseeable future, and probably increase. It is likely that much of this progress will require faster and faster computers, but again in the short/medium term there is no reason why computer power and memory etc won’t continue to increase, even if that just means building bigger and bigger supercomputers.

However, it is notoriously difficult to predict the future, and despite the confidence with which many commentators express their opinions, any long-term predictions should only really be treated as educated guesses. Indeed, we might find some major obstacle to developing human-level AI, or Moore’s law might run out. Or other more extreme things might happen such as nuclear war or a unanimous international agreement that we don’t need to develop AI. My personal opinion is that there is no clear reason why human-level intelligence can’t be developed on a computer, and I would guess that this would happen sometime this century. A survey of relevant experts (presented in Nick Bostrom’s excellent book Superintelligence) mirrors this, with 50% of the people surveyed saying that human-level intelligence will be attained by 2040, and 90% saying it will be attained by 2075.

Even so, no one really knows whether and when human-level AI will be developed. For this reason, I think that a much more important question is not will it be developed, but could it be developed. For me there is a clear answer to this: to the best of our knowledge it is not impossible to develop AI with human-level intelligence. Even if some experts think it is impossible (see below), any rational analysis that takes probability and human biases seriously should take an overview of opinions and therefore give the probability of developing human-level intelligence as greater than zero.

If human-level intelligence is developed then, assuming that at this point we still want to develop AI further, we should expect an intelligence explosion, i.e. a rapid increase in the intelligence of the AI system. The reason for this is that an AI with human-level intelligence (in every department) would also possess human-level abilities to research and develop AI. Then, we only need the AI to be slightly better than humans at developing itself in order to create a self-development loop of increasing ability and speed, leading to an explosion in the AI’s capabilities. Thus, quickly a human-level AI would quickly develop into a superintelligent AI.

Before moving on, I should mention that some think that there are fundamental reasons why human-level intelligence cannot be developed. The main criticism that I know of is based on Roger Penrose’s Gödel-like argument. I’m not an expert on this, and will only mention it briefly, but in short Penrose uses a logical argument to argue that the human mind is non-computable. At least in his book Shadows of the Mind, Penrose seems to largely use this argument to rule out AIs being conscious, rather than ruling out AIs being as intelligent as humans (consciousness and intelligence are completely different, and should not be conflated – I will discuss this in a future post). Even so, there are two reasons why I think his arguments aren’t a major problem: firstly, even if the human mind isn’t computable, then Penrose still thinks it can be explained within the laws of physics. Then, it seems reasonable to assume that eventually we will understand the human brain, and when we do we could in principle construct a computer to replicate the non-computable aspects of the brain. Secondly, as far as I know the aforementioned non-computable tasks are very specific, and are not necessary for most relevant tasks we deem “intelligent”. A future AI not possessing these extra non-computable abilities could still function just as well as humans in many arenas, and would vastly surpass humans in many. Truly human-level intelligent AIs might then not be possible, but for all practical purposes and for most tasks, human-level intelligent is still possible.

To conclude this section, at the very least the development of superintelligent AIs might be possible, and to many it is probably possible. As such, we should take the potential dangers associated with them seriously. If you disagree with me so far, please comment and let me know why!

Why would a superintelligent AI take over and exterminate us?

To demonstrate why a superintelligent AI might cause a threat, Nick Bostrom introduced the paperclip-machine argument. Suppose we have a superintelligent AI, and suppose that we give it the simple and seemingly harmless goal of creating as many paperclips as it can. Quite quickly we can see that this will get out of hand: making paperclips requires resources, such as materials and energy, and if the machine’s only goal is to create paperclips, then it will soon devour the resources of our planet, and continue on into outer space, harnessing all available resources and creating paperclips as it goes. If this scenario were to actually happen, humans would soon see that the paperclip machine is causing trouble, and presumably shut it down. However, remember that the AI is superintelligent – it surpasses humans in all types of intelligence, including social intelligence. It will therefore, very quickly, realise that the humans might want to shut it down, and this would clearly foil its paperclip-making plans, and so it would take every measure to make sure it is not turned off by humans. A good strategy might then be to destroy the humans, and put their atoms and molecules to good use.

This scenario is clearly hypothetical – why would we want a superintelligent AI to just create paperclips?! But we run into the same problems if we give the AI other more “beneficial” goals. We could give the AI the goal of making all humans smile. But how do we define a smile? If it is a specific facial shape, then a logical solution for the AI could be to create a metal structure that can be connected to the face, forcing the shape of a smile. Now to give a less silly example, what about giving a superintelligent AI the goal of eliminating all suffering in the world. Again, this can easily be misinterpreted. One solution to eliminate all suffering is to eliminate all humans, which clearly wasn’t the point. In a similar fashion, the goal of maximising human happiness could be achieved by putting humans’ brains in jars, and pumping them full of serotonin!

From this we can conclude that we need to be extremely careful about what we ask the AI to do. And furthermore, we need to be extremely careful that our commands aren’t misinterpreted. So far, this problem, often called the value alignment problem, is far from being solved. It is a revealing exercise to try to think of a goal, or a set of goals, to give a superintelligent AI; and then to find creative ways that these goals could be misinterpreted, resulting in potential disaster. Anyone interested in this (or anyone skeptical of this argument) should read Nick Bostrom’s book Superintelligence.

Despite the challenges, there are some ideas that seem to be heading in the right direction in specifying appropriate goals for superintelligent AIs. One example, which I think comes from Eliezer Yudkowsky, is to give the AI the goal of first trying to discover what humans would decide we want, if we had infinite time and infinite resources to think about it. And then to implement whatever it thinks humans would want. This still has its problems: it is not clear how we could program such a goal into an AI, and even if we did program such a goal, the AI might run forever and never come to a decision; and clearly not all human wishes align, so it might prioritise the creators’ wishes only, which might be devastating for everyone else.

Even if you are not convinced so far, note that everything above assumed that the creators of the AI will be friendly and want the best for humanity. This might of course not happen, for example if less well-intending governments pumped vast sums of money into AI research. A superintelligent AI would clearly be a powerful, probably even unstoppable, weapon.

Worrying about superintelligent AI taking over the world has only been taken seriously recently, and initially only by a tiny proportion of AI researchers. However, now it is becoming much more mainstream, and has been widely endorsed as a serious problem, e.g. see https://futureoflife.org/ai-principles/. Some researchers and commentators are still sceptical, but as with the development of superintelligence itself, there are no fundamental reasons why we should be certain that we can control a superintelligent AI. Because there are no fundamental reasons, we should, as a species, assume that we might develop AI, and that we might not be able to control it.

The only convincing counter-argument to this that I know of is based on the Fermi paradox. From our understanding of human evolution, and also the evolution of the first life on Earth, the probability that an Earth-like planet could produce intelligent life seems to be high enough so that we are not the only intelligent life forms in our galaxy. Then, in the entire observable universe there would be trillions of intelligent life forms. The probability that we were the 1^st to emerge would be small, but if other intelligent life forms exist then they would presumably develop AI, and if there are no major hurdles to developing superintelligent AI they would have probably done it by now already, in which case we would know about it, because the superintelligent AI would utilise the resources of the observable universe to achieve its goals whether they are beneficial or dangerous. Of course, there are many assumptions here, any which could be false, but one possibility is that superintelligent AI is not so easy to develop after all. But even if this argument seems convincing, it would be unwise to use it (or any other argument for that matter), to rule out the dangers of superintelligent AI with 100% certainty.

A glorious future?

Everything good in our world has been created, directly or indirectly, by intelligence – namely our own intelligence. What, then, would the world look like if we created a superintelligent AI and did manage to control it? It seems reasonable that such a system could, given enough time, solve all of our problems: disease, starvation and war; incompetent governance; other existential risks such as nuclear war, biotechnology, nanotechnology, and global warming; and even more subtle issues like why many of the well-off people in the world – all the people with sufficient food and shelter and friends – can still be miserable, sometimes to the point of depression or suicide. Presumably, a world in which all these problems are solved would be a wonderful place to live in, and therefore in such a world we should be happy to produce many more humans, maybe even as many as possible, to live rich and happy lives. How many of such humans could exist in the future? Bostrom gives various estimates, depending on different assumptions, that range from 10³⁵ to 10⁴³, and even as high as 10⁵⁸ human lives! Given that our successes or failures in AI research over the next hundred years or so might be the critical determinant in whether these lives exist, or whether all humans are wiped out, we might just be at the most important time and place in our universe. I will give Nick Bostrom the final word:

“If we represent all the happiness experienced during one entire such life with a single tear drop of joy, then the happiness of these souls could fill and refill the Earth’s oceans every second, and keep doing so for a hundred billion billion millennia. It is really important that we make sure these truly are tears of joy.”

References: I got most of the inspiration and ideas in this blog from Nick Bostrom’s book Superintelligence; Bostrom and Cirkovic’s Global Catastrophic Risks; and various episodes of Sam Harris’s (fantastic) podcast, in particular those with Max Tegmark, Stuart Russell and David Chalmers https://samharris.org/podcast/.

Quantum mechanics is often presented as being “our most successful theory ever”. Despite 100 years of stringent experimental tests it has never been proved wrong. It has been confirmed to an accuracy of 1 part in 10¹² and has even now been tested in space-based experiments. It underlies much of modern technology, including pretty much the whole information and computing industry. And it has predictive power in an extreme range of scenarios, from the smallest constituents of our universe to a fraction of a second after the Big Bang. For these reasons, it seems reasonable to assume that quantum mechanics is correct, and does not need to be modified, at least not yet.

We can then ask the question: If quantum mechanics is correct, then what does it tell us about the universe? But before answering this, we should probe whether this is a reasonable question in the first place. There is another option: quantum mechanics could just be a tool that is extremely powerful for predicting the outcomes to experiments, and we could say that it’s equations don’t directly tell us about the world itself. To explore this option, we can ask what it would mean for other scientific endeavours. Is the theory of dinosaurs only useful as a tool for predicting what kind of fossils we will dig up in the future, or does it tell about tell us about real creatures that existed in our past? When DNA was first discovered, was the theory of its double-helix structure only useful as an explanation of why a certain x-ray diffraction pattern was observed, or does it tell us about a real molecule that exists in ourselves? And are our theories of stars and galaxies only relevant to explain why we detect certain patterns of light in our observatories, or do they tell us about fantastic objects beyond our solar system? Of course in all these cases we assume that our theories tell us about real objects that exist independent of us. In that case, it seems reasonable to assume that quantum mechanics is telling us about the universe in which we live, about its structure and behaviour and our place in it.

What, then, does quantum mechanics tell us about the structure of the universe? To answer this I will refer to the Schrödinger’s cat thought experiment, which I introduced in more detail in two previous posts (here and here). Briefly, a cat is placed in a sealed box containing a radioactive atom and a vial of cyanide. If the atom decays, the cyanide will be released, killing the cat. It is straightforward in quantum experiments to put the atom into a strange state known as a superposition state, in which it has both decayed, and not decayed, at the same time. Now comes the bizarre prediction of quantum mechanics: if the atom is in a superposition, then the cat will also end up in a superposition of being dead and alive.

It is reasonable to wonder whether the cat can really be dead and alive simultaneously. One option here is to assume that the prediction must be incorrect, and therefore that quantum mechanics is incorrect and must be modified in some way. But, as we have seen above, modifying quantum mechanics should not be taken lightly. Before taking such a drastic step, we should first explore whether the situation can be explained without having to modify quantum mechanics. Yet another option is to say that the equations in quantum mechanics aren’t telling us about the actual state of the cat, and are only a tool that tells us what would happen if we observe the cat. But again, it is a drastic departure from how we normally view science, and should only be used if we can’t make sense of the situation otherwise. I discussed these alternative options here and here.

But despite being wildly counterintuitive (I will return to this later), what is the problem with saying that the cat is dead and alive simultaneously? The problem comes because we know that when we open the box we will either see a dead cat or an alive cat. If the cat was truly dead and alive, shouldn’t we expect to see a weird fuzzy cat when we open the box? The answer is no. Remember we are attempting to use unmodified quantum mechanics directly, so we should use it now to get us out of this riddle. What, then, does quantum mechanics predict we will see? The answer is even more obscure than before: quantum mechanics does not predict that you will see a fuzzy cat, but rather it predicts that you will enter a superposition of seeing the alive cat and seeing the dead cat. Again there seems to be an immediate intuitive objection to this. Surely I cannot be in a superposition – surely I would notice this and be able to tell in some way that I am in this obscure state.

Again we can use quantum mechanics to resolve this, but it turns out that it takes a significant amount of work and effort to do this. Nonetheless, the now well-established theory of decoherence, which I explain in a previous post and in this article, provides the answer. To cut a long story short, decoherence shows that the two parts of the superposition cannot interact or interfere with one another, and for all practical purposes (FAPP) they can never know of the other’s existence. We therefore interpret this superposition state as follows: FAPP there are two versions of you, one who has seen a dead cat, and the other who has seen an alive cat. These two versions cannot interact with each other, and will never be able to know of the other’s existence. When you open the box we say that the universe “branches”. From your perspective you either see the cat dead or alive. But there will be another version of you somewhere out there, on another branch of the universe, who sees the other outcome.

Another reasonable objection here could be to question whether it makes sense to interpret both parts of the superposition as being real. But with some straightforward calculations it can be shown that it is perfectly reasonable to do this: the physics is completely the same whether there is just one version of you or two. It can be shown that both parts of the superposition obey the laws of physics as usual, and essentially live in what appear to be normal universes.

Initially there was just one cat. But after the radioactive decay and cyanide release the universe branched into two “sub-universes”, one containing the dead cat and one containing the alive cat. These sub-universes are called branches, but they are also popularly termed “worlds”, hence the theory being called many worlds theory. They are even called parallel universes, in which case the totality of universes is the “multiverse”. But this terminology can be misleading, because there is really only one world, and one universe, which contains a superposition state of a dead and alive cat. Proponents of this theory prefer to call it “Everettian quantum mechanics”, named after the great Hugh Everett, who first realised that quantum mechanics could be interpreted in this way.

The superposition then spreads out in the following way: Before opening the box there was only one version of you. But on opening the box you join the branches of the superposition state. On one branch the cat is dead and you are observing the dead cat, whereas on the other branch the cat is alive and you are observing it as such. If you then talk to your friend they will also join the branches. On one branch they might see you as relieved that the cat was alive, whereas in the other branch they are consolidating you or berating you for your unethical choice of experiment. As you and your friend interact with more people, and more objects, the branching spreads. In fact, it often only takes a tiny amount of information to be distributed for branching to happen. A single photon that has interacted with you can fly out of the window hitting a nearby building, at which point the building also splits and joins the two branches. This process carries on – as information spreads more and more things join the branches.

I am unsure whether it is known the exact manner by which the branching spreads, and at what rate etc, but the basic theory is now pretty advanced in predicting that, despite the obscure superposition state that quantum mechanics predicts we live in, the normal classical reality emerges and we are completely oblivious to the other branches (e.g. see quantum Darwinism). I should also briefly mention that this branching process happens almost continuously all around us. The fundamental constituents of our bodies – the electrons, quarks, and photons – are constantly splitting into superpositions, and often these superpositions lead to branching in a similar way as in the Schrödinger’s cat thought experiment. For example, in chaotic systems or systems at a critical point small changes in the microscopic initial conditions can lead to vastly different macroscopic final states. Then, if the initial conditions are in a superposition, which they often will be, a branched macroscopic world will emerge from this.

Schrodinger’s cat is dead in one branch of reality, alive in another, reading a newspaper in a third, and on holiday in a fourth! (Image by Joseph Hollis: josephhollis.com.)

I will now address some of the common criticisms against many worlds theory. I invite comments to this blog of more criticisms, and I will try to address these in a future blog. I will try to keep things un-technical, but near the end things might get a little technical in order to give more complete explanations.

Common criticisms

“Many worlds is far too radical.”

On the contrary, the basic formalism of many worlds theory can be seen as our most conservative option! As explained above, the other options are to assume that quantum mechanics is incorrect, and therefore modify it in some way. Or to assume that the equations in quantum mechanics do not tell us about reality. As I argued above, the first option is radical in the sense that it disregards our most successful theory ever, and tries to change it. Whereas the second option goes against pretty much all of our scientific thinking to date. Many worlds theory, on the other hand, takes quantum mechanics seriously, at face value, and we have seen above that such a theory can indeed explain the apparent paradoxes presented in the Schrödinger’s cat thought experiment. The only catch is that the picture of reality predicted by such a theory is completely counterintuitive…

“Many worlds is unintuitive and ridiculous.”

Should intuition be used to rule out theories? It used to be counterintuitive that the world was round, or that the Earth orbits the sun, or that we evolved from monkeys. But our intuition seems to have been refined to encompass these. It is still counterintuitive that all objects around us are largely empty space, or that our bodies are filled with billions of microbes that are essential for our survival, or that our planet is flying through space at millions of mph. The last point bears a resemblance to the explanation of Schrödinger’s cat above: wouldn’t we expect to feel the wind rushing past us if we were flying through space? At first sight maybe yes, but after more careful analysis scientists have made perfect sense of this conundrum.

For me, the examples just given make it very clear that our intuition should not be used to rule out theories. Sure, if our theories predict contradictory things, such as 1+1=1, then we should seriously question our assumptions and calculations. But the picture of reality that many worlds predict is self-consistent and fits perfectly with our observations.

“Many worlds theory unnecessarily invents the existence of almost infinite numbers of worlds.”

Many worlds theory doesn’t invent the existence of other “worlds”. We don’t take the Schrödinger’s cat paradox and say “we can solve this paradox if we invent multiple parallel worlds”. Rather, we take the reasonable assumptions that quantum mechanics is correct and that we can take it literally, and from this we predict the existence of multiple worlds.

“Surely the universe isn’t that big.”

What we thought of as our “universe” used to be much smaller. Ancient Europeans used to believe that the whole world was just Europe, and other cultures thought in similar ways. Then eventually people discovered that the world was actually round and was far bigger than just one continent. Then we discovered that the sun was not just a part of our sky, but rather we were part of a huge solar system. Nowadays, it is assumed that the universe is humongous, maybe infinite, and contains billions of stars. At each stage our picture of the world grew exponentially. Sure, the jump that many worlds theory makes is unimaginably bigger than these other jumps, but it still fits with the pattern that the universe is actually much bigger than we would otherwise expect.

“Many worlds is untestable.”

As discussed above one of the main alternatives to many worlds theory is that we must modify quantum mechanics. This has the obvious perk that it can be tested. Attempts are already being made to see whether gravity collapses the quantum state (thus leaving Schrödinger’s cat either dead or alive, not both). In short, a large mass must be put into a superposition, and it must be sufficiently isolated from the environment so that decoherence doesn’t already lead to the appearance of collapse. Then, because the mass is large, gravity-collapse theories would predict that the mass will spontaneously collapse. Observing this spontaneous collapse would provide strong evidence for this theory.

In many worlds theory, we cannot ever interact with or interfere with other branches of the universe. In this sense, we cannot directly test the existence of these other “worlds”. However, there seems to be double standards in favour of collapsed theories, because if the experiments to test collapsed theories came out negative, then this would rule them out and provide strong evidence that quantum mechanics is correct as it stands. If this pattern continued eventually leading to experiments putting massive objects, or conscious objects, or highly complicated objects such as quantum computers into superposition states, then this would also provide strong evidence that quantum mechanics is correct as it stands. On the other hand, and most importantly, if these experiments proved collapse theories to be correct, then this would directly disprove many worlds theory. Many worlds theory is therefore falsifiable.

One other main alternative to many worlds theory, which was introduced above, is to say that quantum mechanics is just a tool for predicting outcomes to experiments, and it’s equations don’t tell us about reality. This is an alternative philosophy of quantum mechanics, or an alternative interpretation, and in this sense there is no experimental way to distinguish between them. However, in my opinion, if we are eventually able to put massive and maybe even conscious objects into a superposition state, and if repeated experiments prove that decoherence does indeed mean that different branches in many worlds theory cannot ever interact with each other, then these facts would make many worlds theory much more acceptable and plausible. My prediction is that eventually many worlds theory will be the textbook way to interpret quantum mechanics!

Some more-technical objections

The following two issues seem to have been, for a long time, two of the most important technical objections to many worlds theory. However, these seem to have now been solved (at least according to many worlds practitioners!). I won’t go into details, but I’ll try and briefly summarise the objections and their resolutions. However, so that this post doesn’t go on too long I will use a bit of technical jargon at times and assume some more detailed knowledge of quantum mechanics.

Probability in many worlds

Again imagine Schrödinger’s cat just before you open the box. This experiment can be set up so that the amplitudes of the two parts of the superposition (dead and alive cat) are equal. In the usual way quantum mechanics is presented, we say that there would be a 50% chance of seeing the dead cat when the box is opened, and a 50% chance of seeing the alive cat. But in many worlds both outcomes happen. In fact, the Schrödinger equation is deterministic, so many-worlds quantum mechanics is an entirely deterministic theory! So how does the probability come in? In many worlds we say that the probability that “you” end up in the branch with the dead cat is 50%, and similarly for the alive cat. I put “you” in quote marks because there are two versions of you. Just from your perspective you only see one outcome. So in that sense if you see the alive cat, then you will consider this version of you to be the real you! This would be the same for any quantum experiment: the probabilities don’t tell us the probabilities of the different outcomes happening, but rather they tell us the probability that you will end up in a branch with a given outcome.

When the different branches have equal probabilities then the basic idea of probability is somewhat intuitive: initially there was one version of you, but in the end there are two versions of you. Both versions are equally real. Therefore, it makes sense that there will be a 50% chance that the version of you that you experience is the one that sees the alive cat, and so on. However, if the two parts of the superposition have unequal weightings, say 80% alive and 20% dead, then this is no longer intuitive. These probabilities would mean that you have an 80% chance of ending up in the branch with the alive cat. But there are still two versions of you, and they are both real. How can one have a bigger “weight” than the other one? Is one more real than the other?

To resolve this problem, many worlds practitioners (who seemingly largely work at Oxford University!) have provided rigorous proofs to demonstrate that probability does actually make sense in many worlds. They do this in two ways. Firstly, using decision theory, it can be shown that a rational agent should make decisions based on these weightings. For example, if the weightings are 80% and 20%, then a rational agent would be willing to bet £1 at 1/4 odds that they will end up in the 20% branch. Secondly, it can be shown that, if you repeat an experiment many times, in the limit the number of times a certain outcome happens will fit with these weightings. In this sense, you do have an 80% chance of ending up in the branch with an 80% weighting. I acknowledge that this explanation might still feel unsatisfactory, but the maths works!

In addition, many-worlds practitioners will argue that when properly scrutinised, we are far from fully understanding probability anyway (I won’t go through the arguments here, but e.g. watch this). They then argue that certain unanswered questions in classical probability theory can be answered in many worlds theory. In particular, the Born rule can be derived from some simple assumptions (rather than being merely postulated, as it is usually). I’m sure that many readers will be unsatisfied with this brief answer, but the details are beyond the scope of this blog post! Interested readers are referred to David Wallace’s book “The emergent multiverse”.

The preferred basis problem

Historically a serious technical objection to the many worlds theory was the so-called preferred basis problem. I will only give a brief introduction here – I give a much more detailed (and more clear) explanation in this article. In quantum mechanics, there are different bases in which we can represent quantum states. A qubit has two states, which here I will call one and zero. However, we can write these states in a different basis to get plus = one + zero, and minus = one – zero. The two different bases are equally valid, and equally real, and in the lab it is just as easy to prepare the state one as it is to prepare the state plus. Does this same principle hold in the macroscopic world? The answer is no: we only ever see a dead cat or an alive cat. Mathematically it would be equally valid to write these states in the basis plus = dead + alive, and minus = dead – alive. But there is a big difference between these different bases. We know from the Schrödinger’s cat thought experiment that the plus state exhibits branching in many worlds theory. But the dead state does not – the dead state does not branch into a plus branch and a minus branch, as we might expect from the maths. We say that the dead, alive basis is the “preferred basis”. But why is it preferred, and what mechanism makes one basis preferred over and other?

The theory of decoherence now provides rigorous and comprehensive answers to these questions. In short, we can’t just treat the cat as an isolated system, but rather we have to factor in the interaction between the cat and the photons and particles in the box. Once this interaction has been accounted for, it can then be shown that there is one particular basis that remain stable and does not exhibit branching – this is known as the preferred basis. It can then be shown (though not directly) that the preferred basis for the cat is indeed the dead, alive basis. If the cat is prepared in a superposition of these basis states then branching will quickly occur. Whereas if the cat was, for example, alive, then branching would not occur. The same goes for us when we open the box: if the radioactive atom and cyanide were removed from the box, meaning that the cat would always be alive, then we would simply open the box and see the alive cat. No branching would happen. But if the cat is in a superposition then when we open the box branching occurs.

Conclusion

I hope that, at the very least, I have convinced you that many worlds theory is a reasonable and rigorous theory for how we should understand our quantum universe. By taking quantum theory seriously, and literally, we can overcome the paradoxes such as Schrödinger’s cat, and provide a self-consistent picture of our universe. At worst this picture is unintuitive, but at best it paints a fantastically interesting and endlessly fascinating picture of our universe. If you are still not happy with many worlds theory, rest assured that somewhere in the multiverse there is a version of you that is!

Acknowledgements: My general interest and knowledge about many worlds theory comes from David Wallace’s comprehensive book “The emergent multiverse” and Max Tegmark’s popular book “Our mathematical universe”. Specifically for this post, much inspiration and information came from David Wallace’s YouTube talks: https://www.youtube.com/watch?v=2OoRdyn2M9A&t=21s and https://www.youtube.com/watch?v=8turL6Xnf9U&t=2s . I’ve been told that David Deutsch’s books are also excellent!