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:**

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**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 10^{35} to 10^{43}, and even as high as 10^{58} 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 10^{12} 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.

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!

]]>In the __Schrödinger’s cat__ thought experiment, a cat is placed in a box with a device that contains a radioactive atom and a vial of poison. If the atom decays, then the device is designed to release the poison, thus killing the cat. It is now well known that such an atom can be put into a state in which it has decayed, and not decayed simultaneously – this is known as a superposition state. Now, if this system is studied using the central equation in quantum mechanics, the Schrödinger equation, then the following result will be found: if the atom is in a superposition state, then this will lead to the cat being in a superposition state. The cat will be dead and alive simultaneously! Now suppose you open the box – what will you find? The Schrödinger equation again predicts that, if the cat was in a superposition of being dead and alive, then when you open the box *you* will also enter into a superposition. You will be in a superposition of either seeing the dead cat, whilst simultaneously seeing the alive cat.

This clearly does not fit with our experience of the real world. We never see objects in superpositions, and indeed we never seem to experience superpositions ourselves. And while the above experiment is far too challenging to perform using a real cat, conceptually similar experiments have been performed in which an object is put into superposition, and then observed. The result of these experiments fits with our experience and intuition: we never see a superposition state. So what has gone wrong here? Have we misapplied the Schrödinger equation? Is the Schrödinger equation incorrect? The standard resolution, which can be found in most quantum mechanics textbooks, is to introduce the “collapse postulate”: On observation a superposition state collapses, meaning that only one outcome of an observation, or a measurement, is ever observed. I.e. we only ever see the cat as being dead or alive. But the collapse postulate raises as many problems as it solves. What exactly constitutes an observation or measurement? If macroscopic objects are made of quantum particles, *what is so special* about a measuring device or a conscious human observer to cause collapse? (This questions are together often termed *the measurement problem*.)

__Is the Schrödinger equation sufficient to solve the problem?__

Despite how quantum mechanics is often discussed, there is now a widely accepted and carefully studied solution to these problems that utilises the Schrödinger equation alone, and does not have to introduce the troublesome collapse postulate. The solution lies in the theory of decoherence. __Elsewhere__ I give a more thorough introduction to decoherence, in particular in relation to Schrödinger’s cat. But in this post I will try to give a simple and minimal introduction that still captures the main ideas.

Imagine you have a single atom and some cutting-edge experimental equipment capable of putting this atom into a superposition of two locations, A and B. The crucial question here, which is at the heart of decoherence, is: *how do you know it is in a superposition*? If you directly measure the atom then you will either see it at position A, or position B, but not both. To confirm the superposition a more advanced step needs to be taken: we must do an interference experiment. This involves the idea of constructive and destructive interference of waves, which can be seen by throwing two stones into a pond close to one another. The waves coming from one stone interfere with the waves coming from the other stone. If two peaks meet they reinforce one another creating a larger peak, whereas if a peak and trough meet they cancel each other out. Quantum mechanical objects, such as the atom we are trying to interfere, are described by equations known as *wave functions*. As the name suggests, these particles act like waves, and just like the stones in the pond they can demonstrate interference. I’m unsure myself how the exact experiment would work to interfere the two parts of an atom that has been put into a superposition, but by measuring the interference between the two wavefunctions the superposition can indeed be confirmed, and this is now an extremely well measured phenomenon in experiments.

Now suppose you are given two atoms, and you prepare the atoms in the following superposition state: both atoms are in position A, in superposition with both atoms in position B. Again, how can we confirm the superposition? If we directly measure the position of the atoms, then we either find both of them in position A, or both in position B (this is known as an __entangled__ state – the position of the first atom is “entangled” with the position of the second, because we always find them together). Again we do not see, and cannot confirm, the superposition in this way, and we must perform an interference experiment. Now comes the crucial point: the interference experiment must be done on *both atoms simultaneously*, otherwise we will never see an interference pattern. If we just take the first atom, and try and interfere it with itself, then this will not work. (I explain this in more detail __here__.)

We can now return to Schrödinger’s cat. The cat is in a superposition state of being dead and alive, but how can we confirm the superposition? First imagine that the only thing in the box is the cat –it is in a complete vacuum with no air particles or photons or anything. In this case, it is in principle possible to perform an interference experiment with the cat. The dead part of the superposition interferes with the alive part of the superposition, and an interference pattern would be observed, confirming the superposition. This is not practically possible because we would have to interfere every single particle in the cat, and this involves precisely controlling and manipulating every single particle. But according to the laws of physics this is at least in principle possible.

But it is not realistic that the cat could be in a complete vacuum, and no matter how hard we tried there would always be at least a few particles in the box with the cat. These unwanted particles (and photons etc) are often termed *the environment*, and we assume that we do not have control nor access to them. Now again put the cat into a superposition. The cat will inevitably interact with the unwanted particles in the box, and as soon as they interact the cat and unwanted particles will become entangled with one another. Then, if we want to do an interference experiment, we would have to not only interfere all the particles in the cat, *but also all the extra particles and photons in the box*. We would have to precisely control and manipulate all of these particles, but as stated above we are assuming that we cannot control them and cannot access them. Therefore, in this case it is not even in principle possible to do an interference experiment. *We cannot ever confirm that the cat was in a superposition*.

Now what happens when we open the box? As soon as we look at the cat we become entangled with it, and enter into the superposition. The cat is dead and we see a dead cat, *in superposition with* the cat being alive whilst we see an alive cat. But again there will be unwanted particles and photons, and very quickly the cat and ourselves will become entangled with these particles and photons. Again, if we want to confirm that we are in a superposition, we would have to be able to manipulate and control all of these particles and photons, which is clearly not possible. Therefore, again, *we cannot ever confirm that we are in a superposition*. Furthermore, it is likely that some of the photons that have interacted with you will escape from the room through window, flying off to space at the speed of light! In this case, seeing that we can’t travel at the speed of light to collect these photons, it is not even in principle possible to confirm the superposition.

We have now solved the main problems in the Schrödinger’s cat thought experiment. Is the Schrödinger equation wrong? No – we can explain our observations, i.e. that we never see the cat in a superposition, just using the Schrödinger equation. Why do we never see the cat in a superposition? You must do an interference experiment to confirm the superposition, but this is not possible when we factor in the other particles in the box with the cat. The question of “what constitutes a measurement?” has not really been answered yet, but I will address this in a future post in which I defend the many worlds interpretation.

I have not yet fully addressed what happens when you open the box – whether you are really in a superposition, and if so, why you don’t “experience” this superposition. The answer to this really depends on how you *interpret* quantum mechanics, and this is what I will turn to next.

__Many worlds interpretation__

The introduction above to Schrödinger’s cat and decoherence has, in a sense, been written in the language of the many worlds interpretation. In the many worlds interpretation we firstly assume that the Schrödinger equation is sufficient in itself to explain paradoxes such as Schrödinger’s cat, and secondly we assume that quantum mechanics is a theory that tells us about real objects in the real world. The first of these points is justified in the above introduction to decoherence, and nowadays this explanation is widely accepted. The second point is the usual way we interpret science – normally we assume that our equations and theorems are telling us something about a real world that exists independent of ourselves.

These two assumptions might seem quite straightforward, but they lead to quite a radical picture of the world in which we live. For example, in the many worlds interpretation we say that the cat is indeed in a superposition of being dead and alive. There is technically just one cat, but it is dead and alive simultaneously. However, we have seen that the two parts of the superposition cannot ever interfere with each other. Interference is the only way of confirming that an object in is in a superposition, so the dead and alive cats cannot ever know of each other’s existence. Furthermore, the equations of quantum mechanics are such that the future life of the cat (at least the alive one) does not depend on whether the cat is in a superposition or not. Therefore, for all intents and purposes we can think of this as two cats, one dead and one alive. This is where the idea of “many worlds” comes from. For all intents and purposes there are two worlds, one containing an alive cat and one containing a dead cat.

The same idea holds when you open the box. You split into a superposition of seeing a dead cat and seeing an alive cat. But again the two parts of the superposition cannot ever know of the other’s existence, because they would have to interfere with one another to confirm this, and this isn’t possible. Therefore we can again treat this as being two separate worlds, one in which the cat is dead and you are presumably emotionally and morally scarred by the experience, and another in which the cat is alive and you will be relieved.

This picture of the universe is clearly unintuitive, and often people reject many worlds outright and come up with all kinds of criticisms of this interpretation. In my opinion most of the standard criticisms are either ill-founded or result from a lack of understanding of the basic theory, and in a future post I will try to flesh out many worlds theory and provide straightforward responses to many of the criticisms.

__QBism – does the wavefunction represent reality?__

*Before continuing, an important comment is needed. Just before uploading this post I was in contact with Chris Fuchs – one of the founders and main promoters of QBism. To cut a long story short, he said (politely but firmly) that (referring to my previous post) “you *

As introduced above, we can represent quantum mechanical objects using an equation known as a wavefunction. The wavefunction tells us everything we know about this object. For example, we could write down the wavefunction for a single particle in an (equal) superposition of two locations. This wavefunction can then be used to predict what we will see if we perform certain measurements. For example, using the wavefunction we can calculate that, assuming the superposition is equal, if we measure the position of the particle then it will be in position A with 50% probability, or position B with 50% probability. Furthermore, we can use the wavefunction to predict what will happen if we perform an interference experiment. In particular, it will tell us the properties of certain outcomes: it will say that if we perform interference experiment X, then outcome Y will happen with probability Z.

Numerous experiments over the years have confirmed that quantum mechanics is extremely good at correctly predicting outcomes to experiment. But, in a sense, QBism says that *this is all that quantum mechanics is good for*. It says that we should not interpret the wavefunction as describing a real object, and therefore it is meaningless to ask if the cat is really dead and alive simultaneously. We simply cannot know – all we know is the probability of what will happen if we open the box. More specifically, the wavefunction represents our state of knowledge. It tells us what we know, not what exists. This is similar to Bayesian probability theory, in which probabilities this represent our knowledge of the world, not the world itself. For this reason QBism can also be called quantum Bayesianism.

I certainly have some sympathy with QBism. It takes quantum mechanics seriously, and in particular the Schrödinger equation, and does not try to modify the formulae. And it certainly has a strong point: how do we ever really know what exists? The answer is that we observe it, and we perform measurements on it, and we devise clever experiments to perform measurements on the extremes of scale and energy. But until we measure anything, we cannot truly know what it is, and whether it exists. So in this sense QBism is right that quantum mechanics is just a toolbox for predicting experiments.

But is this all quantum mechanics is? Throughout most of human history the goal of science has been to learn more about the world. We do astronomy and astrophysics to learn about stars and galaxies; we smash particles into one another in colliders to learn about what matter is made of; and we do quantum experiments to learn about the weird and wonderful properties of the quantum world. QBism therefore is a radical departure from how we normally treat the scientific endeavour. It is not necessarily the wrong way to interpret quantum mechanics, but Qbists should at least acknowledge that it is an extreme philosophical position.

To take this further, imagine the Schrödinger’s cat thought experiment, but with your friend opening the box rather than yourself. QBism is perfectly good at predicting what your friend will see when they open the box. But, presumably, you believe that your friend exists, and you might be interested in what happens to them when they open the box. QBism cannot tell us this – you can write down the wavefunction for your friend, but this is only a tool for calculating what you will see when you interact with your friend. Many worlds, on the other hand, is perfectly well-equipped to ask questions about your friend. The answer may be disturbing – that they in effect split into two versions – but at least it is a consistent and coherent answer. And this idea can be extended: many worlds theory predicts that almost continuously the world – and therefore your friend – splits into almost infinite parts of a vast superposition, which we can think of as parallel universes.

Would my assumption that my friend exists be incorrect? Perhaps. Maybe in the “real” world it is meaningless to ask about the state of things before we interact with them. But my friend certainly does exist in my head – I can imagine them walking towards the box, opening it, and looking inside. We can then call this world the “imaginary” world. Even though it might not exist outside my mind, I am still interested in what my imaginary friend is doing in this imaginary world. Removing yourself from the picture, now imagine a scene familiar to yourself, such as your house, or your pet, or your favourite sports team. I wonder what they are doing right now? Are there near infinite numbers of them, in near infinite parallel universes? Or is it meaningless to ask what they are doing right now, and only meaningful to think about what happens when you interact with them in some way?

My favourite thing about QBism is this: the wavefunction is normally written using the Greek letter psi, which is often pronounced “sigh”. Ontology is the study of the existence of things, whereas epistemology is concerned with knowledge rather than existence. Therefore, a Qbist is a psi-epistemist. Whereas someone like me who believes in many worlds and therefore that the wavefunction is real, can be termed a psi-ontologist. It deeply troubles me that I am a psi-ontologist (say this sentence out loud to yourself if you don’t get the joke!).

__Collapse theories__

Until reasonably recently it was not fully appreciated that the Schrödinger equation alone can lead to the appearance of collapse. Therefore, to explain why we either see the cat as dead or alive a “collapse postulate” was introduced into quantum mechanics. Initially it was just a postulate, and no explanation was given of how collapse takes place, or what causes it. But this introduces many difficult questions: What causes the collapse? It is usually assumed that a measurement causes collapse: but what is a measurement? Often it is said that a “measuring device”, or even a conscious observer, is what causes the collapse. But if macroscopic objects are made of quantum particles, *what is so special* about a measuring device or a conscious human observer to cause collapse?

Over the years various theories have been introduced to explain collapse with the hope of answering the above questions. Various mechanisms have been proposed: complexity causes collapse – the more complex a system, the more likely it is to collapse; or consciousness itself causes collapse; or gravity causes collapse – the larger the mass, the more likely collapse will occur. These models therefore can explain why Schrödinger’s cat is never seen, or measured, as being in a superposition state.

But now, with the theory of decoherence that I introduced above, we can explain the appearance of collapse without having to add extra postulates into the theory. Collapse theories are therefore unnecessary to explain our observations. So why do they still exist? I have never met anyone who both understands decoherence, and thinks that it is wrong, so collapse would presumably happen in addition to decoherence. And if you are uncomfortable with the conclusion that the cat is in a superposition (many worlds), or that it is meaningless to ask about the state of the cat (QBism), then you can modify quantum mechanics – specifically, modify the Schrödinger equation – so that the state collapses. But for me this seems like a case of changing the science in order to fit our wishes.

This might not be a problem if quantum mechanics was a young and underdeveloped theory. But this is certainly not the case, and the Schrödinger equation itself is responsible for quantum mechanics often being termed “our most successful theory ever”. Do we really want to modify such an equation? Quantum mechanics also works relativistically (i.e. combining it with Einstein’s special relativity), and it has been extended to quantum field theory, which has successfully predicted the Higgs boson. But collapse theories are far from achieving such extensions.

To be fair to gravity-induced-collapse, at some point quantum mechanics, as with any other theory, will be surpassed by some other theory. Quantum mechanics will still be an excellent approximation in many regimes, but in the extremes it will surely break down. But what are these extremes? Potentially the fact that general relativity and quantum mechanics cannot yet fit together gives a clue to this. In this case, might gravity in fact collapse the wavefunction? In my understanding this is at the heart of Roger Penrose’s suggestions to both explain collapse and unify general relativity and quantum mechanics.

For me the main positive to collapse theories is that they are testable. This is especially true for gravity-induced-collapse. If we put bigger and bigger systems into a superposition, while sufficiently isolating them from the environment so that decoherence doesn’t cause the appearance of collapse, then eventually at a certain mass threshold these systems should spontaneously collapse. These experiments should be possible in the relatively near future, and will serve to either confirm this theory, or give extra weight to non-collapse theories such as many worlds.

Consciousness-induced-collapse is in principle testable, but this is far beyond current experiments. To confirm this we would have to put a conscious entity into a superposition. We would have to isolated sufficiently it from the environment so that there is no decoherence, and we would have to be able to control and manipulate every particle in the conscious entity so that we can do an interference experiment. If the consciousness spontaneously collapses, thereby preventing interference, this will be strong evidence that consciousness does induced collapse. The best route to this could be using quantum computers. If we can simulate consciousness on a computer, then we could upload this program to a quantum computer, and subsequently put the consciousness into a superposition. But we don’t even know what consciousness is and such a test is infeasible for now. In addition, I argue __elsewhere__ that if consciousness did cause collapse then the reality this would lead to would be far more bizarre and absurd than even many worlds theory predicts!

__Pilot wave theory__

Einstein famously stated that “God does not play dice”. He simply couldn’t believe that a fundamental theory of nature such as quantum mechanics could really be probabilistic. For example, generally in quantum mechanics we would say that on opening the box containing Schrödinger’s cat it would be random whether the cat is observed as dead or alive (with a certain probability of each). In many worlds theory both outcomes may exist, but it is random whether you end up in the part of the superposition with the dead cat or with the alive cat, so in this sense it is still random. In contrast, theories such as general relativity and Newtonian mechanics are deterministic. For example, if you know all of the positions and velocities of the planets in the solar system, then you can predict with certainty where the planets will be at any given time in the future.

To prevent the randomness of quantum mechanics a “deterministic hidden variable theory” was devised (named Bohmian/De Broglie/pilot wave theory). Taking again the example of the cat, in this theory there are additional variables beyond those in the Schrödinger equation. If we knew the values of all these variables, then we would know with certainty whether the cat will be dead or alive when we open the box. However, these variables are “hidden”, meaning they are fundamentally beyond our measurements and observations. We cannot, and will not, ever be able to determine these values, and therefore quantum mechanics will always appear to be random.

For me this is an even worse case than collapse theories of changing the science so that it more closely fits with our intuition. For protagonists of this theory it is so important that nature must not be random that they are willing to invent an underlying deterministic world that we cannot ever even in principle see. But why should nature be deterministic? In addition, the Schrödinger equation itself is deterministic, so in fact many worlds theory is a deterministic theory. We know with certainty that the cat will be dead and alive. The randomness just comes in when you ask “which universe will I end up in?”. But it is still, from the outside, deterministic.

There are some further complications/criticisms to this theory. John Bell famously showed that, if these hidden variables exist, then they must communicate with one another faster than the speed of light. Furthermore, in a recent __paper__ Renato Renner showed that hidden variable models cannot be self-consistent (although this might not necessarily mean that they are wrong?!).

__Conclusion
__

There are many other interpretations of quantum mechanics, and many more seem to be invented year-on-year. My personal view is that quantum physicists need to stop inventing new interpretations, and consolidate the old ones. Indeed both many worlds and QBism have some features that are unsatisfactory to some and unintuitive to all. But in my understanding *there is nothing fundamentally wrong* with either of these. Sure there are small problems that need to be ironed out, but this is the same for any theory. My personal prediction is that in 100 years from now, if we survive existential risks such as nuclear war or artificial intelligence taking over the world, pretty much every quantum physicist will either be a Qbist, or believe that we live in a fantastic quantum multiverse!

On the other side of the spectrum, scientists are asking seemingly disparate questions within the field of quantum information. How much entanglement do I need to pass on this message? How can I stop my quantum computer from losing its quantum-ness? What’s the best way I can make a quantum superposition?

Now it may seem that there can be no possible link between the answers to these questions. That a steam engine has nothing to do with quantum entanglement, well that is mostly true. However, all of questions are concerned with the same problem, the extractability and conservation of a given resource. On the thermodynamic scale, engineers want to know how heat and work behave. On the quantum scale, theorists and experimentalists alike are interested in defining and conserving the quantum-ness of a given system.

In order to bridge a gap between these fields, we can start by defining a state. Any system, whether it be a quantum or classical, can be described via a state. There are many different ways in which to write down a state, but in order to emphasize the specific resources under consideration I will be writing my states in matrix form.

There are many interesting mathematical and physical motivations for writing states in this way, however I will only be employing two properties of its form: (i) the elements of the matrix correspond to probabilities; (ii) it provides a good pictorial description of the system.

Arguably the most important state in the field of thermodynamics is the thermal state. This is the final state of any interacting thermodynamic system. What do I mean by this? For example, if I was to leave a hot cup of tea in a cold room, they would eventually reach the same temperature as they exchanged heat. This final state of the overall system would be a thermal state.

When writing the thermal state in matrix form, the system orders its state such that the diagonal values of the state become more or less populated depending on their energy. Where the lower energy levels become more populated in comparison to the higher ones.

Given *any* isolated state, if you temporarily attempt to extract work from this state and then let your system relax into whatever state it wants to, if the final state of the system is a thermal state, then you know that you have completely extracted all possible work.

The next obvious question to ask is, what is the opposite of a thermal state? The state from which the most amount of work can be extracted from. This state is called a pure state and be written as:

What makes the pure state so special is the thermodynamic context is that only a single element of the matrix is being populated. It’s as if the components that make up the system have all crowded into the highest energy element possible. Work can then be extracted from this state as the other lesser energy states populate themselves from this one.

Now we have explored the full range of thermodynamic energy states we can now start to think about quantum resources. The quantum resource we will focus on is called quantum coherence. This is a foundational quantum resource that is responsible for a wide range of quantum effects, such as quantum supposition and multipartite entanglement. So how can we possibly grasp any understanding of this complex quantum feature? Well, if you were wondering what happens when the off-diagonal elements are not zero, that’s quantum coherence!

So, the state in thermodynamics whose resource has been fully extracted is the thermal state. What is its equivalent within the resource theory of quantum coherence? It’s called an incoherent state and is written as:

Any state whose elements are entirely concentrated on the diagonal are incoherent states, this includes the thermal state and pure state. Therefore, you know that you have completely extracted all of your available quantum coherence when you end up in an incoherent state.

So what is the state with the maximum amount of extractable coherence, the analogue to the pure state in thermodynamics? It’s called the maximally coherent state and can be written as:

Crucially for the maximally coherent state, every element is identical. As operations are performed on this state that reduce the amount and size of off diagonal elements, the coherence of the state is extracted. This can be repeated until all the coherence is extracted and forms an incoherent state.

So what can we do with all these definitions? Is there some way to bridge the gap between the resources of quantum coherence and extractable work. Well to some degree this is still an active area of research and one with which I’m currently engaged. However, we can at least make a start by attempting to classify the states and attempt to bridge between the resources.

For example, if we order the states from most to least resourceful we produce the following spectrum of states.

Ordering the states in this fashion prompts us to ask some questions.

It appears that coherent states exist past the boundary of what states would normally be considered when extracting work from your thermodynamic system. However, recent work suggests that thermodynamic resources can be extracted from the coherence of a state. Does this mean that the full hierarchy of thermodynamic resource states stretch into the quantum realm?

There are several different classifiers that determine where a state appears on this spectrum of extractable resources. For the part of the spectrum considered in thermodynamics we can compare states via a property called majorisation, which determines if one state can be transformed into another without the input of resources. Interestingly, in coherence resource theory, the property of majorisation is used when considering pure to pure state transformations. Could this be because pure states seem to be the boundary states between the two parts of the spectrum?

This is made more interesting when considering that some of my recent work has developed thermodynamic like relations for the resource of coherence for a pure to pure state transformation. Do thermodynamic relations for coherence resource theory only exist when considering the pure states that exist on the boundary?

It is hoped that the answers to these questions will not only help our fundamental understanding of thermodynamic and quantum theory, but also on the boundary between these two fields (if one exists). So perhaps as we extend our thermodynamic theories further and further into the quantum realm, it may not be too long till your train is powered by the quantum realm after all.

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__Schrödinger’s cat__

One of the founders of quantum mechanics, Erwin Schrödinger, proposed the following thought experiment: A cat is placed in a sealed box with a device that contains a radioactive atom and some poison gas. If the radioactive atom decays, then the device is designed so that it detects the decay of the atom and subsequently releases the poison gas into the box, and this tragically kills the cat. Our intuition says that there are two options here. Either the atom decays and the cat is dead, or the atom does not decay and the cat remains alive. But quantum mechanics tells a different story. In quantum mechanics objects can have more than one property simultaneously, and in particular it is possible to put the atom into a state where it has both decayed, and not decayed, at the same time. But it doesn’t stop there: quantum mechanics also predicts that if the atom has both decayed and not decayed, then this leads to the poison being released, and not released, at the same time. In turn, quantum mechanics predicts that the cat will be dead, and alive, simultaneously!

What do you think would happen if you were to open the box and look at the cat? Would you see the cat as being both dead and alive simultaneously? The answer is of course no – a large object such as a cat has never been seen in such a bizarre state. But why not? Quantum mechanics predicts that the cat can be dead and alive, and quantum mechanics has never been proved wrong. There seems to be a paradox here! But we need not fear, because there are a range of different theories that solve this riddle, and I will introduce some of the main theories below. Bear in mind that none of these theories have yet been proved wrong, and so you are free to choose whichever theory you like…

__Collapse theories__

Perhaps the most intuitive explanation is to say that the description above is not quite correct, and we must add an additional rule that prevents objects such as cats from having multiple properties, such as being dead and alive, simultaneously. In other words, quantum mechanics must be modified slightly, and once this is done it will better fit with our view of reality. So how exactly should we modify quantum mechanics? What should this new rule look like? There are different theories of precisely what this rule is, but they all involve the idea of the quantum state “collapsing”. Using the example above, they say the cat cannot be dead and alive simultaneously, and therefore the state of the cat must “collapse” into being either dead **or** alive, but not both. Now, however the collapse works, we know from experiments that small objects such as atoms *can* have multiple properties simultaneously, so the collapse does not happen at this scale. So what are the main differences between cats and atoms that mean that the cat collapses but the atom doesn’t?

**Gravity causes collapse.** Cats are vastly more massive than atoms. One collapse theory, developed by Roger Penrose and others, exploits this to say that gravity causes collapse. Specifically, the more massive an object is, the more likely it will collapse. This theory says that atoms are small enough so that they can have multiple properties, for example having decayed and not decayed, simultaneously. This is precisely what we see in experiments. However, the cat is so large that, with near certainty, its state will collapse into being either dead or alive.

**Complexity causes collapse.** The main theory of this sort is known as the Ghirardi–Rimini–Weber theory, and it is actually quite similar to gravity causing collapse. It basically says that the more particles an object is made of, the more likely it will collapse. A cat is made of many many particles, and therefore, again with near certainty, its state will collapse into being either dead or alive.

**Consciousness causes collapse.** Now, we don’t actually have a universally agreed upon definition of what consciousness is, and so the theory that consciousness causes collapse is far from being precisely formulated. It of course depends on precisely which creatures (or artificial intelligences!) are said to be conscious. Many would agree that a cat is conscious, and this theory would roughly then say that such a conscious creature cannot have multiple properties simultaneously, and therefore its state will collapse into being either dead or alive. However, if you think that a cat is not conscious, or you replace Schrödinger’s cat with Schrödinger’s microbe (or something else you deem to be not conscious), then this theory would predict that whatever is in the box *is* dead and alive simultaneously. Only when you open the box, and your consciousness interacts with its contents, would the state collapse, and you would be left in either an elated state of seeing an alive creature, or a devastated state of seeing a dead one.

The theory that consciousness causes collapse is perhaps the most compelling, and to some people the most intuitive, explanation of why we never see the cat as being dead and alive simultaneously. However, as I argue here, the picture of reality that this theory predicts is fantastically bizarre and obscure, and far from intuitive!

__Many worlds theory__

The collapse theories introduced above all modify quantum mechanics in some way, and by doing this they can explain why we never see a cat that is simultaneously dead and alive. But is it really necessary to modify quantum mechanics? According to many worlds theory the answer to this is no. However, as explained above, unmodified quantum mechanics predicts that the cat is dead *and* alive, so there is clearly some explaining to do to unify this prediction with our view of reality.

Before opening the box, the cat is dead *and* alive. Technically there is only one cat, which is simultaneously dead and alive. But the great insight of Hugh Everett, who first proposed this theory, was that we should actually treat it as *two cats*, one dead and one alive. Can we really do this? To show that we can, some calculations need to be done, in particular using a framework known as decoherence, but this is too technical to introduce now; see __here__ for an introduction to decoherence in the context of many worlds. The important conclusion from these calculations is that the dead and alive cats can never interact with each other: the alive cat cannot see the dead one, and it can’t smell it nor touch it; as far as it is concerned the dead cat need not exist. For this reason, the usual terminology is that there are two “worlds”, one containing a dead cat and one containing a living cat. This is the idea of “many worlds”. Another way to put it is that there are two parallel universes, with one cat occupying each. But whatever terminology you like to use, the important point is that it is completely consistent within quantum mechanics to say that both cats are equally real, and for all intents and purposes they exist isolated from one another.

What then happens when you open the box? The answer is that you split into two versions of “you”, one that sees the alive cat, and one that sees the dead cat. Again these two versions of you can never interact, and have no way of measuring each other’s existence. They are, for all practical purposes, in separate parallel universes.

Many worlds theory in fact predicts that our reality is almost continuously splitting into multiple parallel universes. In each parallel universe there will be a different version of you. There will be almost infinite versions of you, each going about their day oblivious of all the others. This may seem completely far-fetched, but just because something is not at all intuitive does this mean that it is wrong? It used to be considered absurd that the world is round, or that the universe is vastly larger than our solar system, or that our bodies contain billions of microscopic organisms without which we couldn’t survive.

__QBism – what do our quantum mechanical equations really tell us?__

Are we looking at all this in completely the wrong way? Imagine the state of the cat before the box is opened. Using quantum mechanics, it is in principle possible to write down an equation representing the state of the cat. What would this equation really tell us? In many worlds theory, and indeed in most ways of thinking about quantum mechanics, this equation tells us *what state the cat is in*. Specifically, we are assuming that the cat *does exist*, and that our equation tells us something about it.

But we can take a different perspective of what this equation represents. To see this, we can ask the question: what do we normally *use* this equation for? The answer is that we use this equation to tell us the *probability* that, when we open the box, we will see an alive cat. We cannot use the equation to tell us with certainty whether the cat will be alive or dead – it only ever tells us the probability. For example, it will be possible to set up the thought experiment so that there is a 50% chance of seeing an alive cat once the box is opened, and a 50% chance of seeing a dead cat. Now, over 100 years of experiments have shown that quantum mechanics is extremely good at predicting the probabilities of different events happening in experiments. In fact, as quantum mechanics has never been proved wrong, it is so far perfect at predicting probabilities of outcomes to experiments. Therefore, we know the probability of what will happen when we open the box, and repeating the experiment many times would indeed show that half the time the cat was alive, and half the time the cat was dead.

But what makes us think we know what is happening inside the box before we open it? One way of looking at quantum mechanics, which is often called “QBism”, is to say that our equations do not directly tell us what happens inside the box before we open it. The equations just tell us the probabilities of different events happening. In particular, our equations *don’t* directly tell us that the cat does exist, and that it is both dead and alive simultaneously. The same can be said for all other quantum experiments. For example, when we measure a radioactive atom we can use quantum mechanics to calculate the probability that it will decay. And with today’s simple quantum computers we can calculate the probability that, given a certain input, we will measure a certain output. But our equations do not tell us the state of the atom or the quantum computer before the measurement.

This way of thinking about quantum mechanics has similarities to the question* if a tree falls in the woods with no one around, does it still make a noise?* If we replace the tree with the cat, and the woods with the box, then the QBism answer is that we cannot know anything about the cat before we open the box! Normally, we think of science as telling us something about a real world independent of us, that still exists regardless of our presence in it. QBism takes a different view: quantum mechanics is just a toolbox for predicting probabilities of events.

To many this will seem like a limited view, or perhaps a pessimistic view of the capabilities of science. But how do we really know what happens before we observe/measure anything? The extreme version of this viewpoint says that we can never truly know anything other than our own conscious thoughts. How do we know we aren’t in the matrix? How do we know that the signals entering our brains aren’t just fed into us? The much more conservative version of this view is that a real world *does* exist independent of us, but quantum mechanics doesn’t tell as anything about it. Either way, the riddle of Schrödinger’s cat is no longer a problem: Is the cat really dead and alive before we open the box? The answer is that we do not, and cannot, know. It is a meaningless question!

__“Shut up and calculate”__

Still unsatisfied? Are you not willing to modify a theory that has never been proved wrong? Or believe in almost infinite parallel universes containing almost infinite versions of you? Or is it unsatisfactory to reject the *existence* of things before we measure them? There are some other ways of looking at quantum mechanics which I haven’t mentioned, such as pilot wave theory or relational quantum mechanics, but in my view each of these has significant overlaps with some of those introduced above. Therefore, if you completely reject *all* of the above viewpoints, then maybe you are destined to never be satisfied!

But is this really a problem? Quantum mechanics works, and it works extremely well. It is often stated as being “our most successful theory ever”, owing to the extremely precise predictions of quantum mechanics that have been vindicated, and the vast number of successful experiments over the past 100 or so years. One further viewpoint, then, is that we shouldn’t care whether the cat is dead, or alive, or both. Instead of being distracted by parallel universes and bizarre thought experiments, we should focus on using quantum mechanics better. This is particularly relevant at the moment: the “quantum technology revolution” is making great headways towards fulfilling its promise of transforming future technologies. Quantum cryptography is said to make communication 100% secure; __quantum metrology__ promises to make ultra-precise measurements allowing us to investigate previously-inaccessible phenomena; and quantum computers have the potential to exponentially speed up our computations, thereby revolutionising the whole computing industry. Should people like me therefore stop quibbling about philosophical obscurities, and knuckle down to the real business. Indeed, should we *shut up and calculate*?

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I first came across the *Entanglement* picture about 11 years ago. Nearing the end of my year-long visit to the Centre for Quantum Computation in Cambridge (UK), I was returning to Salerno (Italy) to start working on my PhD thesis. As my dissertation was entitled *Entanglement of Gaussian states*, I decided to include some pictures related to (quantum) entanglement at the beginning of each part. The first few hits for “entanglement” found at the time on Google Images eventually made the cut (check the final version at https://arxiv.org/abs/quant-ph/0702069) However, one of the images in particular captured my imagination – and stole my heart – so much that I chose it for the best spot: Part II, showcasing the bulk of my original results on bipartite entanglement. The image was a relatively low-res photo of a painting entitled, quite aptly, *Entanglement*, and taken from the website of its creator, American artist Pamela (Pam) Ott: http://www.hottr6.com/ott/.

I got so excited about the waviness of the lines, the use of complementary colours, the blurred entwining of two bodies which are manifestly inseparable parts of the same entity, that I decided to send an email to Pam asking whether she had any knowledge of quantum mechanics (as well as for her permission to reproduce the picture in my thesis). This was our first contact. Surprisingly, she replied explaining that she was barely aware of the quantum connotations of the term (see these past blog posts for more about quantum entanglement) and she rather painted *“from her subconscious”*. I was stunned! Judging from that grainy thumbnail, Pam’s painting was, in my opinion, one of the most beautiful and effective ways to showcase the concept of entanglement – perhaps only second to writing actual formulas which, however, do not usually appeal to laypeople as much as art does.

Later in 2007, during a sleepless night as part of my post-doctoral position at Universitat Autonoma de Barcelona, I decided to write more about the potential impact of quantum theory on popular culture, focusing specifically on an introduction to entanglement and its metaphorical connections to human feelings. This was my first foray into public outreach, and the resulting essay, *The social aspects of quantum entanglement*, can be regarded as the forerunner of *Quanta Rei*. Pam’s painting featured prominently in the piece. The essay turned out to be translated and published in a Catalan magazine of popular culture, *Ordint La Trama*, in June 2007, where it made the cover story. For the cover image, Pam kindly sent us a gorgeous high-res photo of *Entanglement*, which has then made its way into many of my talks (both at conferences and public events), and which I later adopted as profile picture for the Facebook page on Quantum Correlations.

By the end of 2008, I was not content with just photos of the painting anymore and asked Pam to buy the real thing. Even though the painting was special to her, she was happy for me of all people to have it. We had even agreed on a price, and she was looking into shipping options, when I got my Lectureship in Nottingham and relocated with my family to the UK. As I was a bit overwhelmed by the move and the new responsibilities at work, and Pam was also moving across different states in the US, our conversation about the painting somehow phased out.

Flash-forward to Spring 2016. The day I got the good news of my promotion to Full Professor, and learned that I would move into a new, bigger office, was the day the craving for *Entanglement* came back with a vengeance. The picture ought to be hanging there on my wall. I tried the old email address I had for Pam, without success. Luckily, thanks to social networks, I was able to reconnect with her and we have been maintaining a regular contact in the past year. Pam continues to paint on a daily basis and I urge you to browse through her virtual gallery on Flickr and Facebook: you’ll find marvellous sketches and paintings, experimenting with a plethora of styles and colours…

We were still discussing about shipping options with Pam, when the occasion for me to visit the US in person materialised this year in the form of a big conference in San Diego (SPIE Optics + Photonics) that took place two weeks ago, in August 2017, and where both my wife and I could present our work. And here we are, just back from a lovely business & leasure trip to California, including a flash visit to Caltech, which was very inspiring especially for my son thanks to the amazing host Spiros Michalakis of *Quantum Frontiers* and MARVEL fame.

The secret main reason for me to go on this trip was to finally bring back the *Entanglement* picture.

I contemplated visiting Pam in person in New Mexico (also taking the chance to do a *Breaking Bad* tour) but that could not fit our schedule unfortunately. Instead, Pam offered to ship the painting to our hotel in San Diego. I could not believe it until I got a big cardboard box waiting for me at the hotel front desk last week. I tried repeatedly to ask Pam for her bank coordinates, but she surprised me once more saying that, after all these years, she just wanted to gift the painting to me! She also shared more about the history of *Entanglement*. In her own words:

“Interestingly, I painted it when I was living in San Diego. I took some art classes at a community college there. That was back in 1999. My teacher convinced me to enter it into the art exhibit at the Del Mar Fair and it won first place, I was pleasantly shocked. I will be honored for you to have it, your work/entanglement has always interested me.”

And so by a twist of fate the painting returned to San Diego, where it was born, and was handed into my care. I felt – and feel – so honoured and grateful! The hardest part was to restrain myself from opening the carefully packed box until we got back to Nottingham.

Finally, two days ago, I came face to face with *Entanglement* in all of its glory. All the tiredness from the long journey disappeared and I immediately ran to my office in the School of Mathematical Sciences and eventually hanged it over my work station.

I am writing this blog post from my office now, raising my eyes to this fantastic and mysterious piece of art, admiring how the colours come to life when caressed by the occasional British summer sunshine. And as it often happens in my personal and professional life, I feel gratified and lucky.

I am sure I will have plenty of chances to further promote the *Entanglement* picture in future work and outreach events. Speaking of what, I am co-organising a Scientific Discussion Meeting on *Foundations of quantum mechanics and their impact on contemporary society* at the Royal Society, London, at the end of this year (December 11-12, 2017). Attendance is free, and we have a line-up of impressive speakers, so if you are reading this post feel free to join us and present a poster if relevant!

I won’t bring the *Entanglement* picture along with me – too much travel for it already – but I will be more than happy to talk about it and in general about the entangling relationship between quantum science and visual arts (see another example in this past blog post).

With special thanks to **Pamela Ott Ingate** for her encouragement and support and the invaluable gift of *Entanglement*.

Follow Pam’s art:

- http://hottr6.com/ott/index.html
- https://www.flickr.com/photos/52431852@N02/
- https://www.facebook.com/PJOtt3

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*I have recently been appointed to the the Editorial Board of Journal of Physics A: Mathematical and Theoretical (in short, J. Phys. A). J. Phys. A is a highly respected journal with a long history of seminal contributions to mathematical and theoretical physics, belonging to the non-profit Institute of Physics (IOP) Publishing family. I have enjoyed publishing in J. Phys. A over the years (including two Topical Reviews) and always experienced a very constructive peer review process; my students love it as well. We had our latest (the first for me) Editorial Board meeting a month ago in Edinburgh and it was a really pleasant and interesting experience, also because I got to spend the week-end there with my family and the weather was surprisingly nice *

*The publishers of the IOP blog JPhys+ interviewed me recently about my career, current research and what it is I find so appealing about the topics I study. The full text of the Q&A interview with Phil Brown, originally appeared here, is copied below. *

**Could you provide us with a brief summary of your career so far?**

I fell in love with quantum mechanics during my undergraduate studies in Physics at the University of Salerno, Italy. I furthered my interests in the field during a PhD also at Salerno, which I completed in 2007. The PhD included a year-long research experience at DAMTP, University of Cambridge which helped me to explore the breadth of quantum information science and its applications. My PhD research focused on the quantification of entanglement in continuous-variable quantum systems, a subject in which I am now regarded as a leading expert. Results from my PhD were summarised in a Topical Review on J. Phys. A which has attracted over 200 citations. After a brief post-doctoral experience at Universitat Autonoma de Barcelona, I moved to the University of Nottingham as a Lecturer in January 2009. Quantum information research at Nottingham has expanded considerably after my appointment. I am now a Professor (from 2016) and head of a research team of 1 junior staff member, 3 postdocs, 7 PhD students, and several long-term visiting scientists, under the umbrella of the newly established Centre for the Mathematical and Theoretical Physics of Quantum Non-Equilibrium Systems.

**You have done a degree of work on Quantum Correlations. What other research areas are of interest to you and what led you to this area of research?**

I have been fascinated by entanglement since first learning about it during my university studies. My recent research, supported by an ERC Starting Grant (2015-2020), has been pioneering in unveiling resources for quantum technology that are more general than entanglement, yet more robust against noise. Such a novel take on quantum correlations, which challenged the two-decade-old separability paradigm, and which I advanced trough several collaborations with theoretical and experimental groups, attracted increasing interest to my work. This progress is presented in another Topical Review on J. Phys. A. I have now contributed key advances to the study of all forms of nonclassical correlations (including entanglement, discord, steering, and nonlocality) and quantum coherence in composite systems. For all these quantum resources, I have proposed faithful measures and discovered operational interpretations, in some cases demonstrated in laboratory.

More generally, I am interested in understanding the elusive boundary between classical and quantum description of the world. This spans from recognising signatures of genuine quantumness in increasingly complex systems, to identifying specific tasks where such resources provide a performance enhancement. These tasks include quantum communication, control, sensing and metrology. I am also working on thermodynamics at the quantum regime, in particular the design and performance optimisation of nanoscale heat engines and refrigerators. Some of my most exotic research plans delve into foundational questions such as how the objectivity of classical information emerges from the subjectivity of quantum observers, and at which rate.

**What kind of problems appeal to you?**

I am fascinated by various types of problems. Sometimes, I find it satisfactory to complete the proof of a rather abstract mathematical theorem, which has nonetheless concrete applications in seemingly unrelated branches of physics. For example, applications of linear algebra and symplectic geometry tools to the characterisation of quantum correlations in harmonic systems are very appealing to me, see e.g. this recent Letter on J. Phys. A and its follow-ups. I am progressively more attracted towards questions challenging the conventional beliefs of quantum information theory, such as where to draw the line between useful and useless resources to demonstrate a quantum supremacy over classical schemes. Sometimes, on the other hand, I like to think of a very concrete problem, such as the performance optimisation of a practical device. In general, whenever a problem admits a neat analytical solution, this makes me particularly happy, but I often resort to numerical explorations in order to guess the solution in the first place. Then, it is usually a challenge for my junior collaborators to prove my intuition right. There have been cases where my intuition failed spectacularly, and investigating such failures turned out to spark a whole new series of interesting questions. This happened e.g. when considering a particular “monogamy” inequality for multipartite entanglement which I had been conjecturing for many years, whose hard-to-find violations eventually revealed a new method to quantify entanglement exactly by simple methods of Euclidean geometry. You can read more about this on my blog.

**What are you currently working on?**

I had a revival of interest on continuous-variable quantum information theory and applications, as I realised there are still a series of unsolved problems where I can contribute, and which rely on interesting mathematical connections that I was not able to reveal during my PhD. In parallel, I am working on the general structure of quantum resource theories, focusing in particular on quantum coherence. The field is rapidly growing (see e.g. my recent review) and is a simple yet important test-bed for both quantum foundations and quantum technologies. I am also focusing on applied and engineering-oriented problems of quantum enhanced imaging and metrology, quantum thermometry and thermodynamics.

**What do you consider to be the most significant problems to be addressed in your field?**

We need to be able to develop a general method to identify resources useful for quantum technologies, and how to exploit them optimally to maximise the efficiency of concrete applications. There are so many different protocols relying on different nonclassical phenomena, yet we still lack a unifying framework. I believe new methods will need to be delivered to address systematically the design and optimisation of new quantum information and communication tasks. The more we understand what makes quantum resources fundamentally different from classical ones, the more we get inspired with effective blueprints to take advantage of them in relevant problems and in realistic scenarios.

**What are the challenges facing researchers in mathematics and theoretical physics?**

I am a physicist in a School of Mathematical Sciences. Sometimes, my publications (e.g. in Physical Review Letters) are perceived as too “physicsy” for the standards of my department. On the other hand, my results usually contain technical bits which are not judged favourably by some higher-impact Physics journal. It is sometimes hard to strike a balance. However, this is a useful challenge for me. I strive quite a lot to craft the presentation of my papers so as not to compromise on rigour on one hand, and to make my results accessible and appealing to a broad audience on the other hand. Dissemination via blog posts and media outlets such as Phys.org and New Scientist help to reach a wider readership, provided the science is not too distorted in such communications. I find it very stimulating to be able to draw from and contribute to both Maths and Physics in my inter-disciplinary field of research. I am sure other researchers in similar crossroads area can be equally challenged and stimulated. In general, the boundary between applied mathematics and theoretical physics is quite blurred anyway, and rightly so.

**Where do you think your research will take you next? Are there any other fields you are interested in exploring?**

I hope I will come closer to understand deep foundational aspects of quantum mechanics, and what are its ultimate limits of applicability. This may possibly give me a glimpse of what lies beyond. I am also very attracted to other emerging technologies, such as additive manufacturing and machine learning. Whether a successful hybridisation and cross-fertilisation of these areas with quantum enhanced technologies is possible in concrete terms, would be a question I’d like to consider in the near future.

**Finally, do you have any advice for young researchers entering the field?**

Do what you love and love what you do. Follow your passion and be sure you find the fun in research, despite all frustrations. Do not get discouraged by the struggles of academia. There are plenty of success stories! Also, do not be afraid to switch fields or evolve your beliefs. As I argued recently in *Nature*, pivoting is pivotal to achieve impact!

In this post, I will give a short explanation of resource theories. We begin by outlining the concept generally, but then focus on two particular examples of resources. The first example considers the resource of money in a fictional bank account, while the second example moves properly into the quantum world with the familiar resource of quantum entanglement.

Let’s consider a physical system that can possess a given resource under investigation. We describe the system with the notion of a *state*, which is a collection of all the information we have about the system. Whenever the system doesn’t have any resource, it is said to be in a *free state*. The free states are one of the main ingredients of the resource theory. Otherwise, when the system has some resource, it is said to be in a *resource state.*

It is also important to describe how we can cause the system to evolve with time. An *operation* describes such an evolution, giving us an output state of the system from the input state. We want to characterise the operations that do not cost us any resource to carry out. These are called the *free operations*, and form the other main ingredient of the resource theory.

As well as identifying the free states of the system, we also want to be able to compare the resource states so that we can understand when the system has more resource. This can be done by using the free operations, which allow us to place the resource states into a hierarchy. Here, one resource state A is said to be higher up in the hierarchy than another resource state B if we can find a free operation that causes the system to evolve from state A to state B. Indeed, since it does not cost us any resource to implement this free operation, we conclude that the system cannot have gained resource and hence state B is not more resourceful than state A.

The shape of the hierarchy can vary for different types of resources. At the bottom of the hierarchy are the free states. However, the top of the hierarchy does not have to be unique: given two resource states it is not always possible to find a free operation transforming from one resource state to the other. This can lead to a hierarchy that has multiple inequivalent branches, where it is not possible to use free operations to transform the system from a state that is in one branch to a state that is in another branch. The figure below shows both a single-branch and a multi-branch hierarchy.

However, the hierarchy is only qualitative. It allows us to compare resource states of the system, but it does not put a number on how resourceful a state of the system is. We can do this by introducing a *measure* of the resource. A resource measure is a function that condenses the complicated hierarchy into a total quantitative ordering, as can be seen in the figure above. It turns out that there is a myriad of possible ways to choose a resource measure. In fact, the different choices of resource measures are a major topic of conversation for quantum information scientists. It is important to highlight two of the basic requirements of a resource measure. First, we require that a measure takes a value of zero whenever the system doesn’t have any resource, i.e. for all the free states. Second, we require that a measure cannot increase when a free operation is applied to the system.

Finally, as we have already discussed at the start, the idea is to use our resource to help us better carry out a particular task. Hence, one of the objectives in constructing a resource theory is to identify a task whose performance is quantitatively given by one of the resource measures.

We now consider the very simple example of the resource of money in a bank account. Here, our physical system is the bank account, while the state of the system is simply the current balance of the account. We can perform operations on the bank account by visiting the bank and withdrawing or depositing funds.

In this example, it is simple to see that there is just one free state – when the balance of the account is zero (let’s assume that this account cannot go overdrawn!). Whenever the balance of the account is positive, the system is in a resource state. Considering now the operations, depositing funds in the bank account requires us to give money to the bank. Hence, the free operations (i.e., the operations that do not consume the resource) are given by all possible withdrawals of funds from the account.

The free operations allow us to see the hierarchy of resource states. Indeed, if we have two possible balances of the account A and B, we know that balance B is less than balance A if we can get from B to A by withdrawing money from the account. Here the hierarchy is very simple, but we see in the next example that this is not always the case.

One obvious choice of a resource measure is given simply by the balance of the account. However, we can also give alternative measures of the resource. For example, we could measure the amount of money in the account in another currency. Alternatively, we could measure the amount of money in the account by how many pizzas we are able to buy with it. This measure allows us to associate the resource with improved performance of a task – namely, the enjoyment of pizza!

In this example we cross into the realm of quantum mechanics and consider the resource of quantum entanglement (discussed in more detail in my previous post as well as our post on monogamy and faithfulness). Our physical system now consists of a collection of quantum objects that we call *qubits*. A qubit, short for a quantum bit, is the quantum analogue of a *bit*, which represents a system that can only exist in two distinct states (think, for example, of the faces of a coin). What makes a qubit “quantum” is that it can actually exist in an infinite number of possible states. However, if we then look at the qubit by observing it in a laboratory, we find that it collapses onto one of two possible states. Have a further investigation of quantum superposition, wave function collapse, and the infamous Schrödinger’s cat thought experiment for more information (as well as our short story* Alice and the Zombie Cat*).

So, let’s suppose that we have a collection of these qubits, which can have the resource of quantum entanglement. Without going into too many details, the qubits are entangled when they can only be viewed as a composite and we cannot describe them individually, see here for an explanation. Hence, the system is in a free state whenever the qubits can be described individually (the exact way to write the free states here is not important, suffice to say that the free states are called *separable states*).

We can also look at the free operations of quantum entanglement. Suppose that we distribute each qubit to a different laboratory that can perform *local operations* on that qubit. Furthermore, we allow each laboratory to communicate their actions to the others through *(classical) communication*, see the figure below. The composition of local operations and classical communication (LOCC) are the free operations of quantum entanglement. Indeed, we do not use any of the resource of entanglement to carry out LOCC.

The LOCC impose a hierarchy on the entangled states of the collection of qubits. The form of the hierarchy depends upon the number of qubits in our system. When there are two qubits, the hierarchy has a single branch with a maximally entangled state given by one of the so-called Bell states. On the other hand, for three qubits there are two separate branches, with the GHZ state and the W state at the top of each branch. The number of branches increases with the number of qubits.

There are many ways to measure the amount of entanglement in our multi-qubit system, this review article and this book both provide an excellent summary and also discuss some of the ways that entanglement can be harnessed in the real world. One of the difficulties of using an entanglement measure is that it can be hard to compute. Nevertheless, for two qubit systems one can use the concurrence, which is a measure of entanglement that is very simple to compute.

*This concludes the post. We finally mention that there are other quantum resources than entanglement, such as quantum coherence. One of the objectives of quantum science is to harness these resources to manufacture technologies that outperform those that are currently available. We are in the early stages of developing these technologies, but one thing is for sure: it is an exciting time to be investigating the quantum!*

*If you have any interesting examples of how resource theories can be applied in everyday life, please write a comment below!*