If you think about Nature carefully you will inevitably reach the conclusion that there are only correlations. Welcome!
‘Well! I’ve often seen a cat without a grin,’ thought Alice, ‘but a grin without a cat! It’s the most curious thing I ever saw in my life!’ – Alice in Wonderland
Pictured: The infamous Cheshire Cat. Its smile is in the top right but the rest of the cat is elsewhere! How does that work?
The Cheshire Cat may seem like the product of a mind descending into madness, but then so is Quantum Mechanics. It will therefore surprise absolutely no one that a quantum version of the Cheshire Cat story exists. The Quantum Cheshire Cat also continues the ongoing fascination that physicists have with cute fluffy cats (see Schrodinger’s cat for another example). I am a dog person myself, but I digress.
Let us break down the Cheshire cat into its parts. The cat has a large toothy grin, you see, but a cat grinning is not the weirdest part. The weirdness comes along when the body of the Cheshire disappears and then reappears somewhere else, but the grin stays behind, floating suspiciously without a body attached.
Inspired by this, Aharanov et al. presented a way to translate this previously fictional weirdness into Quantum Mechanics. Unable to perform experiments on actual cats, they proposed an experiment using photons instead. Photons have an intrinsic property called polarization, which basically tells you the direction that the electromagnetic waves are oscillating. In Aharanov’s experiment, the photons are set up such that they can move along on 2 possible paths – Left and Right, and the polarizations are along 2 possible directions – Horizontal and Vertical. The polarization is a part of the photon, since it doesn’t make much sense to speak of which direction a photon is oscillating in when the photon is not there.
In Aharanov’s experiment, however, it is possible to have the photon in the left path but the polarization on the right! Suppose we implement Aharanov’s experiment and we prepare the photons in exactly the same manner many times in a row. If we were to make a measurement to see if the photon travelled the left path, the detector will always click, 100% of the time. We will therefore conclude that the photon is travelling the left path, not a difficult conclusion to make. However, if we were to make a polarization measurement on the right path, that detector will start giving us clicks! This means there is photon polarization on the right path! We have previously ascertained, with 100% confidence that a photon prepared in the same manner will always travel the left path, so the photon must have performed a “Cheshire Cat”, by moving in the left path, but having its polarization appear on the right.
The above argument is called counterfactual reasoning. Counterfactual reasoning basically refers to arguments made on the following basis: we didn’t do this, but if we did, this would have happened instead. In the previous paragraph, we are measured the polarization on the right path, but suppose we didn’t and measured the path of the photon instead, then we will conclude that the photon is always on the left, but this somehow contradicts our polarization experiment. The apparent weirdness, or paradox, therefore arises because we employed counterfactual logic, since we didn’t (and couldn’t) measure the path of the photon on the same photon we are measuring the polarization, but made our conclusion supposing that we did.
Aharanov et al. of course realized this, and thinking that may be an avenue for criticism, also devised an alternative reasoning based off of weak measurements that does not rely on counterfactual reasoning. I personally don’t think it is a problem per se, but instead perfectly illustrates how our everyday ‘common sense logic’ simply does not apply to quantum mechanics. In any case, it is fun to ponder why counterfactual reasoning does not work in quantum mechanics.
Recently we have worked on a new approach to analyzing correlations. Two parties: Alice and Bob, each generate a binary string (consisting of 1s and 0s). This string may be generated by flipping a coin or by making projective measurements on a quantum state (such as one arm of a Bell pair). Remarkably, we can analyse the bipartite correlations independently of how these strings are generated by looking at how well Alice and Bob’s bit strings can be compressed. Check out the video abstract:
I’d like to pose the following question for this week’s blog: Can you kill a person with a table tennis ball (without eating the ball and dying from poison in the celluloid)? There Well, with the right physics knowledge you might, but I’ll leave you to decide for yourself:
Here’s what a world-class table tennis player can do (which is still pretty cool):
Recently researchers have successfully accelerated electrons using laser pulses . This achievement is far-reaching and will have immediate practical consequences; including methods for building less expensive and smaller devices for medicine and materials science applications. The new method capitalizes on our ability to implement techniques based on commercial near-infrared laser with greater efficiency then any pre-existing technology.
Behind all of this lies a simple idea. A particle’s velocity can be boosted through the interaction between its charge and the electric field component parallel to the particle beam line. Furthermore optimal acceleration conditions, which are experimentally more challenging to meet, dictate that the phase velocity of the accelerating field should always be tuned to the velocity of the relativistic particle.
In this experiment scientist used a micro-fabricated dielectric structure of two gratings (longitudinal cross-section resembles two opposing combs with some space in-between for the electrons to pass through). By preparing the right period for the gratings it was possible to get the diffraction modes of the incident laser pulse to form inside the structure and match the phase resonance conditions.
First results seem to be very promising. This new kind of accelerator offers electron acceleration of 250 MeV/m whereas the standard linear accelerator works at 30 MeV/m. Discoveries utilizing common tools, which promise better solution for practical applications, are always important.
 E. A. Peralta et al., Nature, 27 Sept 2013 (10.1038/nature12664)
Everybody has a fridge. Maybe not an Eskimo, but I have heard that Eskimos actually use fridges to make sure that their foodstuffs are not always completely frozen. I have never personally met an Eskimo to verify this, but anyway everybody has a fridge, and everybody enjoys a nice cold beverage every now and then. Including physicists.
Now, as physicists are wont to do, they start to think: What if you have a quantum beer?
Suppose said beer is a qubit. This is the smallest possible beer since a qubit only has 2 discrete energy states. What is the smallest self contained refrigerator that, in principle, can be constructed?
This was a question that was tackled by Linden et. al. (See link). Their answer: a single atom that can occupy at least 3 discrete energy states. Or alternatively, if you construct your fridge out of qubits, then the answer is 2 qubits which collectively occupy a grand total of 4 possible energy states.
How does it work? Well, technicalities aside, it really is pretty simple. Consider the case where the fridge is made out of 2 qubits. Suppose when a qubit is occupying its lowest energy (ground) state, we label the state 0 and say the qubit is in the state 0. If the qubit is occupying its excited state (the only other energy level), then we say it is in state 1.If you have 3 qubits, then we can describe them compactly in the following way: 000 means all 3 qubits are in the lowest energy state, 001 means only the 3rd qubit is excited, and so on.
All you then need to do is to arrange an interaction between the qubits (or in physics nomenclature, introduce an interacting Hamiltonian) which transfers a 3 qubit state from an energy configuration of 101 to 010. If your quantum beer is the first qubit, then if you start at state 1, you will end in state 0. This means basically that the qubit becomes less energetic and therefore cooler. The problem is that such an interaction between 101<–>010 typically go both ways, which causes the state of your beer to go from 1->0 (cooling) and from 0->1 (heating) with equal measure, messing up the cooling process. This is where a little bit of ingenuity becomes necessary. By putting part of the fridge at a different (higher) temperature, and tweaking the energy levels of the qubits appropriately, you can make the beer and fridge system more likely to find itself in the 101 configuration than the 010 configuration. This means that your beer is more likely to go from 1->0 than from 0->1, and wala! you have a 2 qubit fridge and a nice chill qubit beer.
Everyone has heard about the perfect murder. If you leave no trace, there is nothing that can lead anyone to you. However, if you did something (either bad or good) but no one was watching, is it like you never did it?
Now, think for a moment – is there really no trace? Well, of course YOU know that you DID it, so the trace is inside your head! This information seems safe, but in principle there might exist a sci-fi machinery capable of getting this information from inside your brain, or simply a pair of gentlemen in black suits playing amateur dentists can get it out of you using less sophisticated methods…
However, if you suffered from amnesia, brain damage, or some other unpleasant accident, this information seems to be permanently lost. In this case the question “did it really happened?” seems to be unanswerable.
Recently I rediscovered a paper [PRL 103, 080401 (2009)] (clearly the original event where I first read the paper did exist even if I had forgotten it) that addresses the above question. In particular, it is suggested that events in which entropy is decreasing might actually happen in our universe, but they necessarily leave no trace. Therefore, one cannot observe them and hence we conclude from observable data that entropy can never decrease. Since the job of a physicist is to study things that we do observe, it seems that these strange events are beyond the scope of physics…
One of the cooler things I discovered recently was that when you google search “Conway’s game of life” google automatically begins to play the game of life in the background of the search results page. It is rather mesmerizing,
The game of life is played on a grid of little squares; squares have two configurations “alive” and “dead” which are usually represented by using two different colours. A single game consists of a initial seed or starting configuration and a set of rules which govern how the grid updates:
(1) Any live cell will remain alive if it has 2 or 3 live neighbors; otherwise it will die.
(2) Any dead cell with 3 live neighbors will become alive.
The rules are roughly motivated by physical considerations; if the population density is too low or too high then individual members of a colony will die before reproducing. However three alive cells may reproduce to create a extra adjacent live cell.
The game is extremely popular because the seemingly overly simple rules can generate extremely complicated patterns. Recently it was demonstrated that certain initial conditions of the game of life can give rise to emergent behavior in the sense that we can aggregate large sections of the grid (consisting of multiple cells) and treat this section as an individual “macroscopic cell” which behaves non-trivially. There are no simple set of rules for how these macroscopic cells behave; all update rules continue to apply to the small indivisible cells which constitute them. Nevertheless it is possible to initialize the game of life so that the macroscopic squares effectively appear to evolve according to conditions (1) and (2). There is quite a spellbinding example of this available here. Check it out as it is unbelievable at first sight.
Can you tell apart identical twins easily? Identical twins are quite close to each other so it is hard to figure out who it is at a glimpse. Each twin, however, lives his/her own life, like classical particles having its own trajectory. We can consider the twin paradox in special relativity as an example, assuming that we cannot distinguish identical twins before sending one of them to space. After sending one of twins to space in a high-speed rocket, we can distinguish one twin on the earth from the other twin coming back to the earth, due to gravitational time dilation. In the same sense, we can also distinguish classical particles because they are also in macroscopic dimensions controlled by deterministic mechanics. Now the question is whether we can distinguish quantum particles too? The answer is in one of our papers, http://arxiv.org/abs/1212.5338v3 .
I’m always amazed when I meet scientists who laugh at research into foundations of quantum mechanics. These are usually proponents of Copenhagen school of thought initiated by Niels Bohr whose motto is “Shut up and calculate!”. I’m amazed because I deeply believe that the ultimate goal of science is to understand nature and we are very far away from understanding quantum mechanics, which seems to be a fundamental theory. I can hear now Copenhagen acolytes (those guys are usually bearded and with the air of infinite wisdom hanging around their heads like a saint’s halo) shouting “Speak for yourself! We know the rules of quantum mechanics and they clearly tell us what we need to know.” The only thing I can say in reply to this statement is to quote a guy who probably knew what he was talking about: “So far as the theories of mathematics are about reality, they are not certain; so far as they are certain, they are not about reality.”
So let’s talk about foundations of quantum theory, specifically let’s talk about the reality hypothesis. It was famously incepted into mainstream physics by three guys who dared to think differently – Nathan Rosen, Boris Podolsky and Albert Einstein, EPR for short. EPR wanted to believe that nature is realistic in the sense that all measurable properties of objects in the universe are already out there waiting for us to uncover. Simply, if I open a box and find a hundred dollar banknote inside then the very same note was already there directly before I opened the box. It’s a very reasonable assumption and I’m sure that all of us subscribe to it. If not… you’d better hurry up and check your bank account. To move on with this short essay I need to point out a very important ingredient of the reality hypothesis. Reality does not depend on how you uncover it. Translating this to our one hundred dollar banknote scenario is simple; when we discover a $100 note in the box it is the same banknote regardless of whether we opened it alone, with a bank manager, at gunpoint, in Moscow, New York, and so on. I hope you dig it.
The fact is that quantum mechanics is not realistic in the sense I described above. The banknote will have a different value depending on the circumstances when I open the box. It’s difficult to understand it unless you are one of those bearded Copenhagen guys who say: “Given any circumstances we know how to calculate the probability that the banknote has a certain value. That’s all we need to know because nature forbids us to know more.” Fair enough but I’d like to know more.
Together with one of my colleagues, Pawel Kurzynski, we think we can replace the notion of reality by a new postulate that seems to be more general. This postulate states that some properties of probabilities we observe in nature should also be satisfied by probabilities that we are forbidden to measure. It strongly resembles the reality hypothesis but in fact these two are not equivalent. So far we have managed to prove that the reality hypothesis implies our hypothesis but not the other way round.
The appeal of our hypothesis is that it has a very simple geometrical interpretation and it allows us to derive in a trivial way many known non-contextual inequalities (think Bell and Kochen-Specker inequalities) as well as their monogamies. This is very exciting!
Of course, like in any scientific endeavor we might be on the wrong track. Watch the arxiv for our new paper coming out this Friday.
Suppose you’re on a game show and there are three doors, behind two of which there is a goat, and the remaining one a car. Your objective is to guess which door has the car. You’re allowed two guesses except after the first guess the host of the game show (who knows exactly which door has a goat and which has a car) will open one of the other two doors, always revealing a goat. You’re now given the opportunity to switch doors. The question is what should you do? Switch, or stick with the same door as your first choice (may be because you think the host is trying to play reverse psychology on you). If you’re a probability enthusiast then you’ll know that this is the well-known Monty-Hall problem, and if you’re a not, then (I hope) that you think the answer is very intuitive – namely that it doesn’t make any difference whether you switch doors or not, the probability of choosing the door with the car is 50%. Well, when I first met this problem some years ago I learnt that this intuition is wrong. It completely shattered any confidence I had in probabilities. As a graduate student I learnt what the answer* is and understood the reasoning behind it but the Monty-Hall problem still bothers me somewhat even now. It still bothers me not because I have trouble recalling the solution, but because I found it difficult to resist the temptation to follow the 50/50 intuition on a recent revisit of the problem. It seems that most of us have trouble resisting this temptation (which is why the Monty-Hall problem became famous). To me this means that we just don’t have a robust intuition about probabilities. As the famous polymath John von Neumann said “in mathematics you don’t understand things. You just get used to them.”
*I’m not going to tell you the reason why this is wrong (because it’s readily available on the internet and also because I don’t want to spoil the fun of solving the problem for those who like to challenge).