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!*

Quantum states can exhibit bizarre but powerful properties, such as being in a superposition or containing correlations not possible in classical physics. If these properties can be controlled, then they can be exploited in quantum technologies to dramatically transform computing, enable secure cryptography, and unlock new ways of observing the universe. Quantum optics is a particularly fertile field for testing and developing these technologies – but how exactly can we design a quantum optics experiment to produce useful quantum states of light that can be put to good use? The usual methods involve painstaking calculations, clever insights, and utilising knowledge built up from years of experience and careful reading of previous researchers’ work. But the counter-intuitive nature of the quantum world, whilst enabling disruptive new technologies, can make it particularly challenging to design quantum experiments that can engineer useful states – our usual intuitions can fail us here. Indeed, while the current techniques used by researchers have led to a host of impressive and exciting results, we are far from finding the optimal methods to manipulate and control quantum states.

To overcome this I developed a new technique that instead employs *computers algorithms to design quantum optics experiments* for us^{1}. While computers are not yet creative, and in many tasks can be outsmarted by children, they do have the unique ability to perform millions of calculations per second, and it is this powerful feature of computers that I exploit. Specifically, my algorithm shuffles through different combinations of experimental equipment – such as beam splitters, phase shifters, and non-linear crystals (that “squeeze” the light) – to find arrangements that can produce quantum states of light with specific properties, which can be used for a given task. As with a recent independent project using related techniques by Mario Krenn and Anton Zeilinger^{2}, my computer algorithm found numerous solutions that surpass the previous results in the literature whilst involving surprising experimental arrangements quite different from the human designs.

The picture above is an artist’s impression of the algorithm, named “Tachikoma”. While my first work only found quantum states for making high-precision measurements, future work will find states for a wide range of tasks: highly entangled states, states with a large quantum Fisher information, and the preposterously named zombie cat states and three-headed cat states!

Artwork by Joseph Namara Hollis, __josephhollis.com__.

- P. A. Knott, New Journal of Physics 18, 073033 (2016) http://iopscience.iop.org/article/10.1088/1367-2630/18/7/073033/meta
- Krenn, Malik, Fickler, Lapkiewicz & Zeilinger, Phys. Rev. Lett. 116 (2016) https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.090405

*Happy 2017 from the Quanta Rei team! We close the current year with a guest post from our former member, long-time friend and collaborator Dr Rosario Lo Franco |RLF>*

From the dawn of agriculture until the industrial revolution, all over the world, human beings have been facing the problem of food preservation. We are now quite familiar with many techniques, most of which utilised in our own kitchens, to reach this fundamental goal for our existence on Earth. Efficient and very employed procedures are, for instance: drying, salting, smoking, cooling and freezing. Let us focus on the last one, which works well for a very wide variety of foods. We are aware that, in order to preserve foods for long times by freezing, our freezers and refrigerators must be able to maintain temperatures well under zero Celsius degrees, typically −18° C or below (0° Fahrenheit or below). Air at the poles of our planet would be an extremely efficient freezer for foods, although it is not a very pleasant environment where to live (the average temperatures at North Pole and South Pole are, respectively: 0° C (32° F) and −28.2° C (−18° F) during summer; −40° C (−40° F) and −60° C (−76° F) during winter). There is therefore a continuous technological development in engineering efficient and eco-friendly freezing machines to assure a trustable and lasting food preservation. If someone comes and tells us that it is possible to preserve food by freezing at room temperature we wold not believe them, unless we are in front of Marvel’s Iceman (see picture aside).

Nevertheless, as we highlighted in a recent experiment (http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.117.160402) following a theoretical prediction (http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.210401), there is something very precious at the very fundamental level of matter that is possible to freeze, in principle, even at room temperature: this is * quantum coherence*. Quantum coherence represents the wavelike nature of matter and the essence of quantum parallelism, that is the possibility for a very little system (at atomic scale, nanometers, and below) to be simultaneously in different states. During the last years, quantum coherence has been shown to constitute the primary ingredient enabling the development of quantum technologies for commercial applications to networked secure communication, computing, imaging, sensing, and simulation. The performance of quantum devices in making these tasks greatly supersedes the ordinary classical technology currently available. There is thus great interest in the practical use of quantum coherence other than its fundamental value. But, as often happens when one wants to get an ambitious goal, there are challenging obstacles to be overcome. In particular, one of the biggest problems towards the reliable exploitation of this quantum resource is the unavoidable adverse environmental condition. This is not to be meant as “bad weather” but rather as the destruction of quantum coherence due to the interaction between the quantum system and its surrounding environment, a phenomenon known as

This decoherence is one of the interpretations scientists give of the fact that we do not observe quantum parallelism at macroscopic level. Putting it in a Schrödinger-like fashion, the cat in the box is either alive or dead, it is not alive and dead: the interaction with the environment destroys the possibility of the simultaneous existence of the two “states” for the cat in extremely small and imperceptible times. This is applicable to all macroscopic “classical” objects. At the microscopic level, one can think of an assembly of quantum bits (qubits), which are two-state systems (e.g., atoms, photons, nuclear spins) which can live in their 0 and 1 states at the same time thanks to the quantum mechanical laws. Although quantum coherence between the possible collective states is observable at this scale, it typically lasts for only a fraction of a second before decoherence destroys it. This means that if we want to use quantum coherence for next generation inviolable communications and super-fast computing, we must design and engineer efficient strategies to maintain it as long as possible. Researchers around the world have developed methods to slow down or correct the effects of decoherence but these methods are generally very demanding, requiring external precise controls of the quantum evolution, adjunctive quantum devices and particular structured environments. Differently from these approaches, our study shows a natural mechanism of compound quantum systems to contrast decoherence without any external intervention such that, under suitable conditions, quantum coherence remains unaltered, frozen, during the evolution. Great Scott!, Dr. Emmett “Doc” Brown would say.

Let us now go a little deeper, but shortly, into the story which led us to obtain this result. A story of a group of people meeting in different places at different times to “coherently” discuss the main aspects of the theory and device the experiments. As usual in science, collaboration among people is essential: a sort of human coherence, linking people from Nottingham (UK), São Carlos and Rio de Janeiro (Brazil), Munich (Germany) and Palermo (Italy).

The hard core of the theoretical study and prediction about the phenomenon of frozen quantum coherence was established in Nottingham. Previous works on universal freezing of quantum correlations in compound quantum systems and the introduction of suitable quantifiers of coherence opened the way to the discover that quantum coherence in a system made of an even number of qubits can be in principle maintained constant and equal to its initial value during a non-dissipative evolution. This is thus achievable, once prepared a suitable initial state, whenever there is no energy exchange between the system and the environment, provided that the quantum coherence is measured along a direction which is perpendicular to the direction where the decoherence naturally acts. In fact, the nature of the system-environment interaction creates a given way of action for decoherence, which destroys every coherence it finds along this way, let us say along a *z* axis. To give a pictorial representation, one can see the system made of qubits, where each qubit has two basis states represented by two arrows (spins) pointing up

or down along the *z* direction. The quantum coherence of the system is then observed along the *x* direction, by aligning detectors sensitive to arrows pointing along the *x* axis. And, very importantly, the coherence is predicted to be frozen independently of the measure employed to quantify it: in this sense, the phenomenon is universal. In the picture, you can see Gerardo (right) close to me, along a street of Nottingham, making a call to the labs having this successful “observation gimmick” in mind: our tricky happy faces are quite evident!

It is now the turn of our visit to the labs in Brazil and the planning of the experiment there. It was August 2014 when Gerardo and I went to the Institute of Physics in São Carlos (Universidade de São Paulo) to meet our colleagues there, who soon became great friends. In the picture below, you can see Diogo (right), Gerardo (center) and me (left) smiling during a discussion about the main theoretical aspects of the experiment in nuclear magnetic resonance, who would have reproduced the predictions on freezing coherence, with the background company of the magic music by Vinícius de Moraes and Antônio Carlos Jobim. This picture represents, in my opinion, a very good synthesis of the lovely and funny atmosphere of that time which certainly contributed to the success of the story. The experiment involved people from São Carlos but also from Rio de Janeiro, working at Centro Brasiliero de Pesquisas Físicas (CBPF). Therefore, we spent an entire week in Rio to meet our colleagues at CBPF and follow the work in the lab.

Rio de Janeiro: what an amazing and unique city! I will never forget the landing by airplane there. The pilots introduced Rio to us by a slow spectacular flight above the city, allowing us to immediately appreciate the beauty of houses nested in a lush green nature and vegetation gently fading out into the blue Atlantic ocean, from which small islands and green mountains emerge like sweet fingers greeting the people looking at them. The statue of Christ The Redeemer on the top of Corcovado really gives the feeling of embracing all the city, giving an eye to the Sugarloaf mountain (*P**ã**o de A**çú**car*). It was August, which means summer in Europe, winter in Brazil. But it is universally known that Rio has only two seasons: summer and hell! Therefore, we enjoyed summer at that time, moving from Copacabana beach to downtown samba events. I had the opportunity to go to the *Pedra do Sal* (Rock of Salt), a historical site where samba is played and sang by musicians around a table with the company of hundreds of people, Carioca and tourists as well: amazing, moving and poetical. Thank you Rio for that night! Of course, being those days mainly dedicated to our scientific research, we had the luck to understand that also nucleuses of carbon and hydrogen danced bossa nova subject to nuclear magnetic resonance (NMR) to assembly themselves into the desired quantum states. During our visit to CBPF, where you can see a picture aside with Roberto (right), Gerardo (left) and me (back), the guys of the lab gave us the good news that they would have been able to implement the theoretical method using a setup made of a simple Chloroform sample labelled with Carbon-13, to encode a two-qubit system in ^{1}H (hydrogen) and ^{13}C (carbon) nuclear spins. The systems are naturally affected by dephasing noise (the most common type of non-dissipative environmental noise), so that once the desired initial quantum states are prepared, the freezing of quantum coherence can be automatically observed, without external control.

The German contribution then enters the game with the aim to go further and prove the occurrence of this counterintuitive phenomenon in larger quantum systems. Thanks to our “globetrotter” researcher Isabela, who traveled a lot to connect all the involved people by an effective “human coherence”, the lab at the Technische Universität München was successful in increasing the number of qubits from two to four. This four-qubit system was a heteronuclear sample developed in Steffen Glaser’s group, manipulated by a prototype NMR probe. In particular, the four qubits were encoded in ^{1}H, ^{13}C, ^{19}F (fluorine) and ^{31}P (phosphorus) nuclear spins. Both Brazilian and German setups demonstrated long-lived quantum coherence involving room-temperature liquid-state NMR quantum simulators. Thus, freezing coherence at room temperature is possible, opening up further research on exploiting this resource and understanding its possible role in biological complexes! Thank you Isabela, Alexandre, Tom, Marco, Raimund, Roberto, Ivan, Steffen, Eduardo, Diogo and Gerardo for the great collaboration.

I would like to conclude this story by pointing out the following message: it is often by changing the viewpoint that striking results may surprisingly show up. They are there, Nature is kind of willing to bring them to human knowledge, but one needs to find the right perspective to see them! This concept may be illustrated by a picture I made during my visit at the wonderful Iguazù Falls, at the border among Brazil, Argentina and Paraguay: the beauty of Nature is just there to be seen and appreciated by the proper sensitive eyes!

*This is a guest post by Dr Rosario Lo Franco |RLF>*

When young Tudor visited the West Indies for the first time, he was delighted by the warm Caribbean weather. He may have been sunbathing or walking on the beach when the idea struck him like a lightning. It was a new business model, completely overlooked, and extremely profitable: Why not cut ice in Boston, ship it to the tropics and sell it to local restaurants? They could start selling chilled drinks, or even ice-cream. Most people there had never seen ice before. *They would go crazy about it! *

I can imagine young Tudor considering with excitement the feasibility of his idea. Ice was free. At least in Boston. And there was plenty. One only needed to cut it in blocks. Ships were affordable at the time and, to keep the ice from melting, one could insulate it with sawdust, which was also essentially free. Believe it or not, nobody had thought about large scale commercial ice ventures before. Back then, ice was only used in small quantities wherever it was naturally available. But the idea of getting people to actually pay for ice was simply revolutionary.

And so, the 10th of February of 1806, the brig *Favourite* departed from Charleston headed to Martinique with 130 tons of ice in her hold. Not surprisingly, the venture was an absolute disaster: the ice that didn’t melt during the three-week journey to St. Pierre, melted soon after arrival, since there were no storage facilities in the island at the time. Although he did manage to sell some ice, poor Frederic lost better than $3500 dollars in this first frustrated attempt.

But he didn’t give up. His determination to succeed against all odds was almost heartbreaking. Tudor had “ice houses” built in Martinique and Havana at great expense and started experimenting with different kinds of insulation to improve the efficiency of his business. In spite of his efforts, he kept accumulating debt, and even spent some time in debtors prison between 1812 and 1813. And yet, he kept trying.

As he started getting profits from his sales, his ambition grew out of proportion: He harnessed horses with metal blades to cut ice in greater amounts and teamed up with a business partner to start shipping ice to India.

*(Yes, my friends, you read well. This guy was seriously shipping ice from Boston to Calcutta)*

And guess what? He succeeded. By the 1850s, Frederic Tudor, a.k.a. *“the Ice King”*, had built an ice empire, shipping over 150 000 tons of ice per year to South America, India, Persia and the Caribbean. Tudor died a millionaire, in 1864, at the age of 80.

Crucially, besides making a fortune, the Ice King managed to change the everyday life of people around the globe by creating a new *need*. Larger amounts of ice were stored every winter and transported by boat or train to big cities, where its use for food preservation, medicine or manufacturing of chemicals became widespread. For instance, by the time Tudor died, *lager* had superseded all other types of beer in Germany.

Of course, ice had its detractors too. They say that in Vermont preachers warned in their sermons against “the abominations of sucking soda” as late as 1890, and even laws were passed in some towns prohibiting selling it on Sundays. Around that time, any accidentally frozen food was immediately trashed and the regulations on cold-storage warehouses were extremely restrictive, supposedly in the interest of public health.

Back in 1834, when Tudor was expanding his trade to India, a guy called Jacob Perkins invented the *refrigerator* or “artificial ice-making machine”. Perkin’s patent sketched a primitive hand-operated compression fridge with very limited usability. The idea of artificial refrigeration didn’t really take off until *much* later. It was in 1860 when Ferdinand Carré patented the absorption refrigerator in the US. This ingenious invention, powered by heat, instead of manpower, did find practical applications: During the Civil War, it supplied the Confederates with ice, which they could not get anymore from the North. By the 1890s, steam-powered compression refrigerators and absorption cooling systems were produced for industrial use, although these were still humongous machines weighing from tens to hundreds of tonnes.

Even then, some of the best-informed engineers of the time laughed at the idea of refrigerating machines replacing natural ice, let alone the concept of *domestic refrigerators*. But just like those who laughed at Frederic Tudor when his *Favourite* cleared customs for Martinique in 1806, they were simply wrong.

The history of refrigeration teaches us two important lessons: The first one is that the biggest impediment for technological progress is often the very opposition of *people*. It takes a lot of time and determination to convince scientists, businessmen and the general public of the interest, profitability and practical uses of new technologies and ideas. The second lesson is that businessmen play a starring role in technological progress, side by side with scientists and engineers. Their support and initiative is vital to make technological breakthroughs happen. Sadly, this is very rarely acknowledged.

As you all know, artificial ice eventually won the war against the once thriving ice-shipping empire of Frederic Tudor. Furthermore, small-scale domestic fridges did made their way into our households in the early 20th century.

The first domestic refrigerators were noisy and dangerous, though. All refrigerants used at the time were poisonous, and accidents happened quite often. They say that when reading in the press the tragic news about the death of an entire family, poisoned by the gases of their own household refrigerator, Albert Einstein got very upset. This happened in the mid 1920s, when Einstein was visiting his good friend Leo Szilard. These two theoretical physicists, better known for their works on quantum mechanics, relativity, or nuclear energy generation, embarked themselves in a seven-year collaboration to design safer refrigerators. They produced a total of 45 patents and even managed to *build* several working prototypes of the most ingenious cooling machines that one could possibly imagine.

But that’s another story.

Maybe next time.

]]>*“When two systems enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems*

*separate again, then they can no longer be described in the same way as before. I would not call that one but rather the characteristic trait of quantum mechanics,the one that enforces its entire departure from classic lines of thought. By the interaction the two systems have become entangled”*

*[E. Schroedinger, 1935]*

Being a physicist or a mathematician and living in the normal world is not a simple task. This is not only about physical aspect, conventions, or appearances.

This is also about the weird sensation that you prove before answering the question “W*hat do you do in your life?*” Generally speaking, saying you studied maths or physics leads to the answer “*Wow! You’re very smart*“, that in general is how the conversation ends. However there are more brave people who go further and deeper in the discussion asking you what is your research on.

Sometimes it happened to me that I had to explain to my – non-physicist – friends what entanglement is. This is how usually this thing goes:

– *I’m studying a type of correlation between quantum systems called entanglement.*

– *Sounds interesting! Can you tell me more?* –

– *Well, basically you let two very small systems interact and it can happen that they end up being correlated in such a way that even if they separate again you can’t describe them separately anymore.*–

… and here comes the awkward part…

– *Oooh! So it’s like they fell in love!*–

Since I’m not a very romantic person, the first time I heard this reaction my face looked like this:

But in time I learned not only to take things less seriously but also that sometimes concepts that you would never relate, explain each other more easily (especially to people that are completely unfamiliar with one of the two subjects).

The first time I came in contact with the concept of “*monogamy of entanglement*” it was reading B. Terhal paper celebrating C. H. Bennett’s legacy on quantum information theory [B. Terhal, *Is entanglement monogamous*? IBM J. Res. Dev. 48, 71 (2004)]. Monogamy of entanglement: it makes sense, no? If entanglement can be compered to love, then we should also question its monogamy. So let me explain you what monogamy of entanglement is.

Suppose that you have three systems. Let me address them with *A*, *B* and *C*, for simplicity. Furthermore, suppose *A* and *B* are maximally entangled. It turns out that each of these two systems, say *A* specifically, can’t share any entanglement with *C*. If we want to maintain the romantic vision of entanglement as love, it makes perfect sense to call this property *monogamy of entanglement*: it can immediately become the story of a woman, Alice, and a man, Bob, that are so in love with each other that for them is impossible to find room for anyone else.

Now we can move this concept further and ask more questions. For instance, Alice and Bob may be not so keen on spending all their time always together and with no one else and maybe they would like to meet other people. This is how they meet Charlie. Anyway, being so in love with each other, Alice and Bob cannot dedicate too much time, energies and attentions to someone else, let’s say Charlie.

This is how we arrive to a more interesting concept of monogamy of entanglement. In 2000 it was proved that if two qubits (elementary quantum systems) are entangled, the entanglement that they can share with a third part is bounded [V. Coffman, J. Kundu, and W. K. Wootters, *Distributed entanglement* Phys. Rev. A 61, 052306 (2000)]. In other words the entanglement that *A* shares with *B* summed with the one that *A* shares with *C* can’t exceed the entanglement that *A* shares with *B* and *C* seen as a whole system. The work of Coffman, Kundu and Wootters caught what seemed to be an essential feature of quantum entanglement and since then many other works followed, trying to investigate deeper this property in many directions. In particular, from the rough description that I gave you there are at least two possible directions that are not so difficult to see:

- Study what happens enlarging the number of subsystems;
- Study what happens enlarging the dimension of the single subsystems.

But maybe the experts among you, readers, have noticed that I move forward a crucial information: what measure did they use to quantify the entanglement? Indeed, there’re a lot of mathematical functions that can be addressed as entanglement measures and each of them is able to catch different aspects of this quantum feature. In their paper, Coffman, Kundu and Wootters proved the inequality shown in the previous picture (that from now on I’ll call CKW inequality) using as measure the (squared) *concurrence* [S. Hill and W. K. Wootters *Entanglement of a pair of quantum bits* Phys. Rev. Lett. 78, 1997; W. K. Wootters, *Entanglement of formation of an arbitrary state of two qubits* Phys. Rev. Lett. 80, 1998]. This means that there is a third possible direction in which one can try to investigate further:

3. Study what happens changing the employed entanglement measure.

Here it comes one of the main problems regarding CKW inequality. Indeed, it can be shown that its validity is not universal, but rather depends on the specific choice of the measure. In fact, useful entanglement monotones, such as the *entanglement of formation* or the *distillable entanglement* do not directly obey the inequality.

Anyway monogamy seems to be a key feature of entanglement as hinted, for instance, apart for the nice parallelism with love, by the s*hareability problem* that states that it is not possible to create a symmetric extension of an entangled state of two parties, for infinitely many parties.

So we are led to raise the question:** Should any valid entanglement measure be monogamous** in a CKW-like sense? In particular, given an entanglement measure *E, *should it satisfy an inequality like

with *f* a suitable function that doesn’t make everything trivial? Clearly we would like to have *f *as general as possible meaning that we would like it to be dimension independent, for instance.

The first step in order to answer our question is to define what are the properties that we would like the entanglement measure to satisfy in order to be considered valid. Here I will skim through some technicalities, but let me just say that we would like our entanglement measure to be additive, in the sense that if you bring two copies of the same quantum state the total entanglement you get is twice that of a single copy, and to respect the geometry of quantum states, in the sense of assigning a high value to states which stay far away from the subset of unentangled states no matter their dimension (one such example is the so-called fully antisymmetric state in arbitrary dimension, which has a constant distance from unentangled states). This last requirement can be considered a kind of* faithfulness *of the measure. (Now you see where I’m going, don’t you?) But before going further and before you start thinking that I’m a cheater, let me tell you that these properties have not been chosen in such a way that there is no possible *E *satisfying them. Indeed, important measures of entanglement such as the *regularization of the relative entropy of entanglement* and the *entanglement cost* satisfy them.

As maybe you’re expecting, the interesting thing in this discussion is that it can be proved that a suitable *f* for *E *satisfying the properties that we required, including the non-dimensional dependence (or faithfulness), can’t exist (what a surprise!). Indeed, what can be shown is that enlarging enough the dimension of the system we can always find a state for which the entanglement of part *A* with *B *and* C* taken together is effectively the same as the entanglement that *A* shares with *B *and* C* individually.

This leads us to the starting title and directly to the end of the story. The moral is that even if monogamy and faithfulness are properties that we would like entanglement (maybe also love!) to be endowed with, they’re not compatible to each other at all (is it also true with love?), at least within the weird quantum world.

…Ending titles…

[based on a real story:

C. Lancien, S. Di Martino, M. Huber, M. Piani, G. Adesso, A. Winter

Phys. Rev. Lett. 117, 060501 (2016)]

Ehm… If you’re wondering, the answer is yes: there is a closing scene after the ending titles.

There is a way to escape the monogamy vs faithfulness problem: you can decide to give up the universality of *f * regarding the dimension of the system, thus preserving both monogamy and faithfulness.

*Would you rather care for monogamy, faithfulness or dimensions?*