# Why do they say love when they mean entanglement?

Distance can be so painful. We all have experienced having our beloved ones far away, either in time or space. I can only say that physical pain is milder.

Entanglement was born as the denial of distance. In the thirties, when quantum mechanics was still a child, Schrödinger was astonished to consider that quantum particles could, somehow, keep in touch even when they are separated long distances. Just because they interacted in the past, and they keep the connection.

Let us consider particles which are so dumb that they can only learn how to answer one question, and only with “yes” or “no”. All other questions are answered randomly. Let us put a couple of these particles in strong interaction, designed so that they will always provide opposite answers to any question formulated to them both. Since they can’t memorize a big list of questions, what they do is the following: the first one to be asked answers randomly, and the second answers the opposite as the first.

But now, to give the interesting twist, consider that the particles are separated a long distance. Still they can’t memorize but one question. Nonetheless, when you ask the same question to them both, they give opposite answers! Don’t call it love, call it entanglement!

By the way, what we described is the Einstein-Podolski-Rosen (EPR) paradox for spin 1/2 particles. Can we use it to send information along large distances? No. But I will leave the reader to think why.

So, these entangled pairs have some kind of connection, which Einstein called a “spooky action at a distance”. It is used extensively by scientists in order to design quantum communication and quantum computers. But I will talk about that some other day. I want to focus on the distance stuff. Entanglement seems to be the denial of distance. Is this true?

Is it rare to find entangled quantum particles? Not at all! Our electrons are strongly entangled to their neighbors, in the same atom or in nearby atoms, thus making up chemical bonds. Typically, these entanglement bonds are of short distance. You may get any block of matter and ask “how many entanglement bonds does this block have with  the rest of the universe”? The answer is normally proportional to its surface area. The reason is that electrons that live deep inside the block only have entanglement bonds to other electrons inside the block. Only the “surface” electrons are entangled to the exterior world. This is called the area law.

A soup of entangled pairs.

But the area law is not always true. Many quantum states, some natural and some engineered, have long distance entanglement bonds. Imagine a blocks in which all the electrons are entangled to a partner which is outside. Life inside that block is weird. Each electron is paired with another one which is out of your reach. So they answer questions in a weird way, which seems totally crazy to you. They are not crazy, they are in love… sorry, they are entangled to other guys which do not live nearby. The system seems random to you. Physicists consider temperature to be the most relevant source of randomness. So, for a scientist living inside such a weird block… a sensible interpretation is that, simply, it is hot. Hot. Really? Because their lovers are far away. Waw. The  metaphor really pays for itself!

But now the twist comes again! Some recent ideas, on which we are working at IFT, suggest that we should look at the relation between entanglement and distance the other way round. You may have heard that the universe is curved. Really. And that curvature is related to the gravitational pull. But you may not have heard of Einstein-Rosen (ER) bridges. If spacetime can be bent, maybe it can be cut and pasted. Why not? And then we can make shortcuts, connections between far-away places. You want to travel from Madrid to Vladivostok really fast? No worries! An ER bridge can do the trick. And this is the conjecture, which was put forward by the argentinian physicist Juan Maldacena: what if EPR=ER? What if we take seriously the area law, and decree that two entangled particles are always nearby? Indeed, this is the case for most of the time. What about the exceptions? If the particles of a given entangled pair seem to be distant to you it is because they are connected through an ER bridge, which, unfortunately, you can’t see. Thus, every time we see distant entangled pairs, perhaps we are just noticing… spacetime curvature at a quantum level.

If Einstein-Rosen = Einstein-Podolski-Rosen, then, is Podolski equal to one?

You may have dozens of objections. For example, doesn’t curvature of spacetime require mass? We do not have a full fledged quantum theory of gravity, so we don’t really know the answers. It helps solve some old problems, such as the black hole information paradox (which we will discuss some other day). But, most of all… it is really beautiful and suggestive.

So, maybe, distance does not exist at all. Maybe you should just get entangled with your beloved ones. But remember… entanglement is monogamous!

Image by RomaniM

To know more:

Quanta magazine article on EPR=ER.

Entry on entanglement at the Stanford Encyclopedia of Philosophy

Schrödinger’s cat and quantum computers.

Original, in Spanish, published at madri+d, at the blog of the Instituto de Física Teórica (UAM-CSIC): http://www.madrimasd.org/blogs/fisicateorica/2015/10/22/103/

# From Schrödinger’s cat to quantum computers (II)

From analog computers to quantum computers

In this second part of the talk, we move on to discuss quantum computers. They are fashionable, they’re cute. You can boast in bars that you work in quantum computers and you’ll get free beers. But… are they, really, more powerful than their classical counterparts? I have devised a way to explain what is the main difference, and why they might well be more powerful. And it uses an old and forgotten concept: that of analog computer.

What is an analog computer? It is a machine designed to solve a certain computational problem, using physics. For example, the Antikythera mechanism. It is a device which was found in a sunk Greek ship from the Hellenistic times. It has some gears which, when spinning, represented the motion of planets, and helped predict their positions in the sky. In a certain way, the astrolabe is also an analog computer.

The Antikythera mechanism, an early analogue computer.

But let us give a simpler example. Imagine that you must order a large set of numbers, from lowest to highest. You can design an “ordering computer”, in the following way: get some spaghetti and cut each of them to a length corresponding to one of the numbers. Then, you take the full bunch and hit the table with it, flattening its bottom. Now, you only have to pick the spaghetti in order: first, second, third…

Another cute example is the problem of finding the most distant cities in a roadmap. I give to you a list of cities, and the distances those which are linked by a road. An analog computer can be built with a long thread. We cut it to pieces representing the roads joining
each pair of cities, and them we tie them in a way resembling the full roadmap. The knots, of course, represent the cities. Now we pick the roadmap by one of the knots, and let the rest hang freely. We look at the knot which lies the lowest. Now we pick it, and let the rest hang freely. In a few iterations we converge to a cycle between to knots. Those are the most distant cities in the roadmap.

And another one! Consider a 2D map on which we are given the positions of a few cities. We are asked to design the road network of minimal length which joins them all. Sometimes it will be convenient to create crossroads in the middle of nowhere. This is called the Steiner tree problem, and it is considered a “hard” problem. There is a way to solve it fast with an analog computer: we get a board and put nails or long pins on it representing the cities. Now we immerse the full board into soapy water and take it out very slowly. A soap film will have developed between the nails, with some “crossroads”, joining all pins. If we do it slowly enough, the film will have the minimum energy, which means that the total roadmap will have the minimal length. This idea is important: we have converted our computational problem into one that Nature can solve by “minimizing the energy”.

Experiments by Dutta and coworkers, http://arxiv.org/abs/0806.1340

Our paradigm problem to solve will be the spin glass problem. It sounds like a technical problem, but I have a very nice way to explain it: how to combine your goals in life and be happy. Well, we all want things which simply do not go together easily: work success, health, true love, going out with the buddies, children, etc. Let us represent each of those goals as a small circle, and give a weight to each of them. Now I draw “connection” lines among the goals, which can be positive, if they reinforce each other, or negative, if they are opposite. Each line has a “strength”. So, for example, “having children” is highly incompatible with “going out with the buddies”, but it is strongly compatible with “finding true love”. Which is the subset of those goals I should focus on in order to maximize my happiness?

Each node is a “target”, blue links mean “compatible”, and red means “incompatible”.

I will tell you how to solve this problem, which is truly hard, using an analog computer. We put atoms in each node, and we make “arrow up” mean “focus on this goal”, while “arrow down” will mean “leave it”. Now, experimental physicists, which are very clever guys, they know how to “lay the cables” so that the energy is minimized when the system represents the maximum happiness. Cool.

No, not so cool. The problem is the “false minima”. Imagine that you’re exploring the bottom of the ocean and you reach a very deep trench. How can we know that it is, indeed, the deepest one? Most of the time, the analog computer I just described will get stuck in the first trench it finds. And, believe me, there are many. It’s just the story of my life: I know I can do much better, but… I would have to change so many things, and intermediate situations are terrible. But today I feel brave, I really want to know how to be happy.

Quantum mechanics comes to our help. Remember that atoms are quantum, and that their arrows can point in some other directions, not just up or down. So I put my analog computer, made of real (quantum) atoms inside a giant magnet, which forces all spins to point rightwards. Remember: $\left|\rightarrow\right> = \left|\uparrow\right> + \left|\downarrow\right>$, so now each one has 50% probabilities of pointing up and pointing down, like Schrödinger’s cat. Maximal uncertainty. I know nothing about what to do. Let us lower slowly the power of the magnet. Nature always wants to minimize the energy, so we pass through some complex intermediate states, highly “entangled”, in which some atoms decide early which position to choose. They are the “easy atoms”, for which no competition with other atoms exist. The “difficult atoms” are the ones in which you have more doubts, and they stay in a “catty” state for longer time. When, finally, the power of the magnet has come to zero, all atoms must have made up their minds. We only have to read the solution, which is the optimal happiness one.

Sure? It all depends on the speed at which we have turned the magnet off. If I am greedy and do it fast, I will ruin the experiment, and reach any “false minimum” of the energy. So, a false maximum of the happiness. All this scheme is called “adiabatic quantum computation”. For physicists, “adiabatic” means “very slow”.

How slow should I go? Well, this is funny. Apparently, most of the time it’s not very important. But there is a critical moment, a certain “phase transition” point, when the entanglement between the atoms is maximal. Then, it is crucial to advance really slowly. As an analogy, think that you have to take a sleeping baby from the living room to his cradle. There is always a fatidic moment, when you have to open the damned doorknob. If you are unlucky, you may have more than one. But, for sure, at least you’ll have one.

And, what if we’re so clever that we have left all doors open? That’s what worries us physicists most. There is a conjecture, that there is some kind of “cosmic censorship”, that will impose a closed door in the path of every difficult problem. Nature might be evil, and has put obstacles to the possibility of solving difficult problems too fast. It would be a new limit: the unsurmountable speed of light, the unstoppable increment of entropy… and now that? It is worth to pay attention: the next years will be full of surprises.

This is the second part of a lecture I originally delivered in the streets of Madrid for the “Uni en la calle”, to protest the budget cuts in education and science in Spain, in March 9, 2013. And, later at a nice high school in Móstoles.

# From Schrödinger’s cat to quantum computers (I)

From Schrödinger’s cat to entangled cats

If you are also fans of The Big Bang Theory, you will be aware of Penny and Sheldon’s discussions about Schrödinger’s cat. Penny wants to know whether she should hook up with Leonard, and Sheldon tells her that, in 1935, Erwin Schrödinger designed a mental experiment in which a cat was put inside a closed box with a vial of poison which can be opened at random times. You may not know whether the cat is alive or dead until you open it. Penny thinks that the lesson is that she should try, and only then she will know. But, really, Sheldon only wanted to get rid of her. The question, as all human interaction, was completely irrelevant to him.

I have a board. If you like boards, this is my board.

Although I love the series, the explanation about Schrödinger’s cat is lame. You put a cat in a box and a vial of poison. There’s 50% chance that the vial opens and the cat dies. According to our intuition, the real state of the system is one of them: alive-cat or dead-cat. Since we don’t know which one it is, we represent our knowledge with probabilities:

But that’s not quantum mechanics! Quantum mechanics is far more weird, and tell us that the cat may be alive and dead at the same time. We represent it this way:

(That notation, $\left|X\right>$, is called a “ket”… yes, we physicists are very fond of funny notations.) If, while in that state, you open the box, the cat is forced to choose. With 50% probabilities, it becomes an alive-cat, and with 50% a dead-cat. But then, how is it different from before!?  Because the “alive-and-dead-cat” is a new “catty state” that we may represent this way:

and which has different properties.

Well, with cats this doesn’t really work. We tried, but they move a lot, and miaow, and scratch. We better try with atoms, which are far more peaceful. Most atoms behave like small magnets, and their magnetization can be thought of as a small arrow, called “spin”, pointing in any direction, something like this:

We have our (cat-like) atom in a closed box, with its little arrow pointing in any direction. With cats, you may ask: “are you alive or dead?”, and it gives you an answer. With atoms you may ask, for example: “is your little-arrow (spin) pointing up or down?” Of course, not only for the vertical direction. You just pick up any direction and ask, but let’s say that the vertical direction is clear enough. So, you may have the atom in state $\left|\uparrow\right>$, and the answer will be “up”, or $\left|\downarrow\right>$ and the answer will be “down”. But what happens if you mix them? You can have the state $\left|\uparrow\right> + \left|\downarrow\right>$. Then when you ask “Is your little-arrow pointing up or down?”, the atom chooses $\left|\uparrow\right>$ with 50% probability and $\left|\downarrow\right>$ with 50%.

(By the way, if someone is thinking of erotic analogies, flash news: we physicists have already thought of all of them.)

But I told you that $\left|\uparrow\right> + \left|\downarrow\right>$ is more than just 50% up and 50% down. Let’s change direction. Now, instead of asking about up or down, we ask “is your little-arrow (spin) pointing rightwards or leftwards?” The atom answers “rightwards” with certainty. 100% probability!! So… that was the point!! It answered randomly when asked about up or down, because it was pointing to the right!! Is there any way to prepare the atom so it points always leftwards? Yeah, we write $\left|\uparrow\right> - \left|\downarrow\right>$. And the same happens if you ask the other way round: if you have $\left|\uparrow\right>$ and ask “are you pointing left or right?”, it will answer randomly. Wrong question, random answer.

But let’s come back to cats. We can go beyond Schrödinger and put two killer cats in the same box. They hate each other, and only one will survive. The quantum state can be written as

but… it could be the other way round! It might be

Classically, we would have 50% of each, But, in quantum mechanics, we can have the state

I put both signs because both are possible. It depends on the cat breed, I think.

But, I insist, that’s hard to do with cats. Do it at your own risk. With atoms, it’s a whole different story. We may prepare atoms such that their little-arrows (spins) point, for sure, in opposite directions. Let’s say that they are in the state

$\left| \uparrow\downarrow \right> - \left | \downarrow\uparrow \right>$

This state suffers from what we call entanglement. And very weird things happen to it. That was studied by Einstein and some of his buddies, called Podolsky and Rosen, in 1935 (also) (yeah, good year), when they showed that we could do the following. Take the box containing both atoms and split it in half, making sure that a single atom stays in each half-box. Now, take one of the boxes very far away. When you ask one of the atoms “are you pointing up or down?”, you don’t know what the answer is going to be, because it chooses randomly. If we do it with cats, you don’t know if the box you’ve kept contains the dead or the alive cat. But let’s assume that you get the answer “up”. Then we know what will the other atom reply when asked whether its arrow  points up or down. It will say “down”.

The surprise comes when you ask the atom: “is your arrow pointing left or right?” Its answer will also come randomly, 50% right and 50% left. I am not going to justify that, just believe me. But if you ask the same question to the other atom, no matter how far away it is, its answer will be the opposite!!! You may ask about any direction, and both atoms will give you opposite answers. The question that we may ponder is, of course… how does the second atom know what was measured on the first? Apparently, entangled atoms hold a bond that, like good loves and good hates, survives distance.

This is the first part of a lecture I delivered in the street in Madrid, as a part of the “Uni en la calle” program to protest the budget cuts in education and science in Spain, on March 9, 2013. I have delivered it also at Manuela Malasaña high school. Thanks to you all, guys!

# Quantum particle near an event horizon

As the next episode in our series about the Unruh effect (it gets hot when you accelerate), here can watch a video I have prepared which depicts how a quantum particle behaves near an event horizon.

So, what are we watching? The left and right panels show the spin-down and spin-up wavefunctions for a massless Dirac particle (a massless electron), initially at rest in Rindler spacetime. Colors correspond to phase. Because of the principle of equivalence, there are two alternate physical interpretations:

• You are moving with constant acceleration rightwards. At time t=0 you drop a Dirac particle. It seems to move leftwards, just because you leave it behind. When it gets far away, it slows down. This is due to relativistic time-dilation.
• There is a uniform gravitational field pointing leftwards. That’s why the Dirac particle accelerates in that direction. As it falls, it slows down. This is due to gravitational redshift.

Of course, the interference pattern which develops at the center is just quantum mechanics, nothing else. But when the particle reaches the edges of the box (top, bottom and right), new interference patterns appear which are spureous to our problem. That’s just the handicap of a finite-size simulation.

Nice, ein? This was work we developed at ICFO, Barcelona, along with Maciej Lewenstein, Alessio Celi and Jarek Korbicz. I have just showed it as a premiere during the Quantum gases meeting at CSIC in Madrid.

# It’s hot when I accelerate!

Let us discuss one of the most intriguing predictions of theoretical physics. Picture yourself moving through empty space with fixed acceleration, carrying along a particle detector. Despite the fact that space is empty, your detector will click sometimes. The number of clicks will increase if you accelerate further, and stop completely if you bring your acceleration to zero. It is called Unruh effect, and was predicted in 1976.

That’s weird, isn’t it? Well, we have not even scratched the surface of weirdness!

So, more weirdness. The particles will be detected at random times, and will have random energies. But, if you plot how many particles you get at each energy, you’ll get a thermal plot. I mean: the same plot that you would get from a thermal bath of particles at a given temperature T. And what is that temperature?

$T = \hbar a / 2\pi c$

That is called the Unruh temperature. So nice! All those universal constants… and an unexpected link between acceleration and temperature. How deep is this? We will try to uncover that.

In our previous Physics Napkin we discussed the geometry of spacetime felt by an accelerated observer: Rindler geometry. Take a look at that before jumping into this new stuff.

Has this been proved in the laboratory?

No, not at all. In fact, I am working, with my ICFO friends, in a proposal for a quantum simulation. But that’s another story, I will hold it for the next post.

So, if we have not seen it (yet), how sure are we that it is real? How far-fetched is the theory behind it? Is all this quantum gravity?

Good question! No, we don’t have any good theory of quantum gravity (I’m sorry, string theoreticians, it’s true). It’s a very clear conclusion from theories which have been thoroughly checked: quantum field theory and fixed-background general relativity. With fixed background I mean that the curvature of spacetime does not change.

Detecting particles where there were none… where does the energy come from?

From the force which keeps you accelerated! That’s true: whoever is pushing you would feel a certain drag, because some of the energy is being wasted in a creation of particles.

It’s hot when I accelerate!! Ayayay!!!

I see $\hbar$ appeared in the formula for the Unruh temperature. Is it a purely quantum phenomenon?

Yes, although there is a wave-like explanation to (most of) it. Whenever you move with respect to a wave source with constant speed, you will see its frequency Doppler-shifted. If you move with acceleration, the frequency will change in time. This change of frequency in time causes makes you lose track of phase, and really observe a mixture of frequencies. If you multiply frequencies by hbar, you get energies, and the result is just a thermal (Bose-Einstein) distribution!

But, really… is it quantum or not?

Yes. What is a particle? What is a vacuum? A vacuum is just the quantum state for matter which has the minimum energy, the ground state. Particles are excitations above it. All observers are equipped with a Hamiltonian, which is just a certain “way to measure energies”. Special relativity implies that all inertial observers must see the same vacuum. If the quantum state has minimal energy for an observer at rest, it will have minimal energy for all of them. But, what happens to non-inertial observers? They are equipped with a Hamiltonian, a way to measure energies, which is full of weird inertial forces and garbage. It’s no big wonder that, when they measure the energy of the vacuum, they find it’s not minimal. And, whenever it’s not minimal, it means that it’s full of particles. Yet… why a thermal distribution?

Is all this related to quantum information?

Short story: yes. As we explained in the previous post, an accelerated observer will always see an horizon appear behind him. Everything behind the horizon is lost to him, can not affect him, he can not affect it. There is a net loss of information about the system. This loss can be described as randomness, which can be read as thermal.

Long story. In quantum mechanics we distinguish two types of quantum states: pure and mixed. A pure quantum state is maximally determined, the uncertainty in its measurements is completely unavoidable. Now imagine a machine that can generate quantum systems at two possible pure states A and B, choosing which one to generate by tossing a coin which is hidden to you. The quantum system is now said to be in a mixed state: it can be in any two pure states, with certain probabilities. The system is correlated with the coin: if you could observe the coin, you would reduce your uncertainty about the quantum state.

The true vacuum, as measured by inertial observers, is a pure state. Although it is devoid of particles, it can not be said to be simple in any sense. Instead, it contains lots of correlations between different points of space. Those correlations, being purely quantum, are called entanglement. But, besides that, they are quite similar to the correlations between the quantum state and the coin.

When the horizon appears to the accelerated observer, some of those correlations are lost forever. Simply, because some points are gone forever. Your vacuum, therefore, will be in a mixed state as long as you do not have access to those points, i.e.: while the acceleration continues.

Where do we physicists use to find mixed states? In systems at a finite temperature. Each possible pure state gets a probability which depends on the quotient between its energy and the temperature. The thermal bath plays the role of a hidden coin. So, after all, it was not so strange that the vacuum, as measured by the accelerated observer, is seen as a thermal state.

Thermal dependence with position

As we explained in the previous post, the acceleration of different points in the reference frame of the (accelerated) observer are different. They increase as you approach the horizon, and become infinite there. That means that it will be hotter near the horizon, infinitely hotter, in fact.

After our explanation regarding the loss of correlations with points behind the horizon, it is not hard to understand why the Unruh effect is stronger near it. Those are the points which are more strongly correlated with the lost points.

But from a thermodynamic point of view, it is very strange to think that different points of space have different temperatures. Shouldn’t they tend to equilibrate?

No. In general relativity, in curved spacetime we learn that a system can be perfectly at thermal equilibrium with different local temperatures. Consider the space surrounding a heavy planet. Let us say that particles near the surface at at a given temperature. Some of them will escape to the outer regions, but they will lose energy in order to do so, so they will reach colder. Thus, in equilibrium systems, the temperature is proportional to the strength of gravity… again, acceleration. Everything seems to come together nicely.

Hawking predicted that, if you stand at rest near a black hole, you will detect a thermal bath of particles, and it will get hotter as you approach the event horizon. Is that weird or not? To us, not any more. Because in order to remain at rest near a black hole, you need a strong supporting force behind your feet. You feel a strong acceleration, which is… your weight. The way to feel no acceleration is just to fall freely. And, in that case, you would detect no Hawking radiation at all. So, Hawking radiation is just a particular case of Unruh effect.

There is the feeling in the theoretical physics community that the Unruh effect is, somehow, more fundamental than it seems. This relation between thermal effects and acceleration sounds so strange, yet everything falls into its place so easily, from so many different points of view. It’s the basis of the so-called black hole information paradox, which we will discuss some other day. There have been several attempts to take Unruh quite seriously and determine a new physical theory, typically a quantum gravity theory, out of it. The most famous may be the case of Verlinde’s entropic gravity. But that’s enough for today, isn’t it?

For references, see: Crispino et al., “The Unruh effect and its applications”.

I’ll deliver a talk about our proposal for a quantum simulator of the Unruh effect in Madrid, CSIC, C/ Serrano 123, on Monday 14th, at 12:20. You are all very welcome to come and discuss!

# How many dimensions did you say?

My friends and colleagues from ICFO, Alessio Celi and Maciej Lewenstein (along with O. Boada and J.I. Latorre), have just published a surprising article in Physical Review Letters, which appears in its Synopsis. What is the big deal?  They propose a route to simulate the behaviour of quantum matter in extra dimensions. The idea is extremely simple once it has been understood. But let me start by telling you what is the framework, I mean: what do I mean by simulation.

Consider that you would like to design a new material, which you want to have some nice properties, such as superconductivity, or a given response to magnetic fields… whatever. Most of these properties are given by the quantum behaviour of electrons inside the crystal. The problem is that the behaviour of interacting electrons in a given system is very hard to predict theoretically, using either pencil and paper calculations or huge supercomputers. What to do, then?

When aeronautical engineers design a new airplane, they do some complex calculations. After that, since they can not rely completely on them, they make a model plane and test it in a wind tunnel. They perform a controlled simulation. If the solution to their equations
coincides with the results of the simulation, then they feel confident about the airplane, and the manufacture procedure begins.

Quantum simulators follow the same idea as the wind tunnel and the model airplane. Set up many laser beams, making up a 3D lattice. The lattice spacing will be much larger than in crystalline solids, more than one μm. Now, instead of electrons, we put some ultra-cold atoms. But, I hear you say, atoms are not elementary particles, unlike electrons. There is a nice response to that: anything is an elementary particle until you hit them hard enough! In other words: atoms behave totally like elementary particles if the temperature and the interaction energy is low enough. If they have total spin 1/2, then the atoms are fermionic and behave much like electrons.

So, you have all the elements. Now, let us check a possible design for a material with some concrete properties: set up your optical lattice, put some ultra-cold atoms in there and see. The best part is that if you do not strike oil at the first attempt, you can always change your parameters almost on-the-fly and try again: tune the lasers, heavier atoms… whatever.

Now that we know what a quantum simulator is, let us focus on the novel part: the work of my colleagues. Many speculative theories in physics require the existence of extra-dimensions. If they exist, then their extension must be really small not to appear in ordinary experiments. I do not mean that those theories should be taken seriously, only that we might desire to find out what would be the implications!

Imagine that we prepare our optical lattice and leave our atoms inside. Atoms jump from a cell to the next tunneling through the laser beam. Now, consider atoms that can be in N different internal states, which differ, for example, in the nuclear spin direction. So to speak, N atomic flavours which are nearly indistinguishable. Label the internal atomic states from 1 to $N$, and arrange things so that atoms can only move from state $i$ to state $i+1$ or $i-1$. Now, by tuning up the laser intensities, we can make this movement in internal state to be exactly as movement in any other direction!

The image shows in blue the atoms with flavour 1, and in pink those with flavour 2. An atom at site $d$ can jump up, right, back… but it can also change flavour. And that jump would correspond to a movement in the fourth dimension. Of course, the extension of this fourth dimension is extremely reduced if we have only two flavours. In general, we will not be able to achieve huge sizes, but this is not a problem since, as we stated, the extra dimension, if it exists, must be extremely small.

For example, we can arrange a single atom in a given cell, with a given internal state, and let it evolve freely. After some time, it will be in another cell and with another internal state. This internal state will mark how much it has moved in the extra-dimension.

V.I. Arnold, one of the great masters, once said that mathematics is the part of physics where experiments are cheap. Well, the cost of the mathematical experiment must always be compared to the cost of the real one. Using an expensive supercomputer to follow the behaviour of all the atoms of a stone as it falls to the ground does not seem to be a huge saving. But using ultracold atoms in an optical lattice to simulate 4D space qualifies much better… most of all because we are not aware of any other experimental setup! :)

# Qubism

Scientists tend to be very visual people. We love to understand through pictures. About one year ago, we had one of those ideas which remind you why it’s so fun to be a theoretical physicist… Simple and deep. The idea was about how to represent quantum many-body wavefunctions in pictures. Speaking very coarsely, the high complexity of the wavefunction maps into fractality of the final image.

So, more slowly. As you know, bit can take only two values: 0 and 1. A qubit is a quantum bit, which can be in any linear combination of 0 and 1, like Schrödinger’s cat, which we denote by $|0\rangle$ and $|1\rangle$. In other terms: a qubit is represented by two complex numbers: $|\Psi\rangle = \alpha |0\rangle + \beta |1\rangle$. If you have two qubits, the basic states are four: 00, 01, 10 and 11, so we get

$|\Psi\rangle = \alpha_{00} |00\rangle + \alpha_{01} |01\rangle + \alpha_{10}|10\rangle + \alpha_{11}|11\rangle$

If you add one qubit, the number of parameters doubles. For N qubits, you need $2^N$ parameters in order to specify completely the state! The task of representing those values in a picture in a meaningful way seems hopeless… Our idea is to start with a square and divide it in four quadrants. Each quadrant will be filled with a color associated with the corresponding parameter.

What if we get a second pair of qubits? Then we move to “level-2”: we split each quadrant into four parts, again, and label them according to the values of the new qubits. We can go as deeply as we want. The thermodynamical limit $N\to\infty$ corresponds to the continuum limit.

The full description of the algorithm is in this paper from arXiv, and we have launched a webpage to publish the source code to generate the qubistic images. So, the rest of this blog entry will be just a collection of pictures with some random comments…

This is the ground state of the Heisenberg hamiltonian for $N=12$ qubits. It is an antiferromagnetic system, which favours neighbouring qubits to be opposite (0-1 or 1-0). The main diagonal structures are linked to what we call a spin liquid.

These four pics correspond to the so-called half-filling Dicke states: systems in which half the qubits are 0 and the other half 1… but you do not know which are which! The four pics show the sequence as you increase the number of qubits: 8, 10, 12 and 14.

This one is the AKLT state for N=10 qu-trits (each can be in three states: -1, 0 or 1). It has some nice hidden order, known as the Haldane phase. The order shows itself quite nicely in its self-similarity.

This one is the Ising model in a transverse field undergoing a quantum phase transition… but the careful reader must have realized that it is not fitting in a square any more! Indeed, it is plotted using a different technique, mapping into triangles. Cute, ein?

But I have not mentioned its most amazing properties. The mysterious quantum entanglement can be visualized from the figures. This property of quantum systems is a strong form of correlation, much stronger than any classical system might achieve.

So, if you want to learn more, browse the paper or visit this webpage, although it is still under construction…

With warm acknowledgments to my coauthors: Piotr Midgał, Maciej Lewenstein (ICFO), Miguel I. Berganza and Germán Sierra (IFT), and also to Silvia N. Santalla and Daniel Peralta.