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).

Searching for relations between different physical processes is one of the most successful ways of doing science. Usually these relations are not only surprising but also useful as they offer us a different approach to certain problems. In statistical mechanics such a relation was showed by Christopher Jarzynski [1] who found an equality connecting equilibrium information with non-equilibrium measurements for finite classical systems.

From thermodynamics we know that in the case of quasi-static or, in other words, processes which take an infinitely long time the work done on the system is equal to the difference of Helmholtz free energy of the system. However, for finite time processes some proportion of the work can be dissipated and so in general the total work may surpass the difference of Helmholtz free energy.  Jarzynski’s result shows it still possible to find the value of that difference from finite time measurements which actually makes the whole thing accessible experimentally [2].

To give you some taste how it is done, Jarzynski simply uses the special method of averaging the exponential function of work. This approach provides better weights for different work trajectories to get at the end the mean value of the exponent of dissipated work equaled to one.  It occurs that from Jarzynski’s equality it is easy to derive the fluctuation-dissipation relation as well as the already mentioned inequality between average work and the difference of the system’s free energy.

One interesting question recently asked by researchers was whether there are quantum analogues of this result.  Reported positive answers only prove the general importance of Jarzynski’s equality.

[1] C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).

[2] J. Liphard et al., Science 296, 1832 (2002).

In Douglas Adam’s Hitchhikers guide to the Galaxy, the author describes a race of hyper-intelligent beings who built a super-computer whose purpose is to compute the Answer to the Ultimate Question of Life, the Universe, and Everything. The computer, christened Deep Thought by its architects, famously gave the answer to be 42. The creators now have a huge problem: they have the answer, but what does the answer really mean?

Now, I am not writing this because I have secret aspirations of being a philosopher. In fact, I tend to run away in uncontrolled panic as soon as a philosopher steps into my peripheral vision. No, the real reason why I thought of Hitchhiker’s Guide is because some scientists right now have a similar problem right now. Except we are not hyper-intelligent beings.  And the computer is not really that super.

On May 2011, a company called D-Wave Systems announced a device called the D-Wave One, declaring it the world’s first commercially available quantum computer. Quantum computing has long been touted as one of the next big revolution in computing technology.  The basic promise of a quantum computer is easy to understand: Problems that may take years or decades using a classical computer may potentially be solved in hours or days using a quantum computer. A closer look will tell you that this does not necessarily apply to every problem you would like to solve, but hey, nobody ever reads the fine print anyway.

In any case, D-Wave may claim that they are selling a quantum computer, but can we really know for sure it is really quantum? If you are purchasing a multimillion dollar device, it kind of makes sense to do some checks to make sure you are not blowing all that cash on snake oil. As it stands, the ‘quantum computer’ in question does give out an answer if you give it the right kinds of questions, but what does that answer really mean? That is what scientists like Sergio Boixo et al.  (arXiv:1304.4595) are trying to find out. So far the results appear to support D-Wave’s claims in that it does indeed appear to be quantum, but skepticism remains. Still, ignoring the controversy a little bit, it is perhaps interesting to discuss a little bit about the science behind how the D-Wave One is supposed to work.

The basic idea behind D-Wave’s system is Quantum Annealing. Annealing takes its name from a process with the same name in metallurgy. When you cast steel, a sudden temperature change can lead to internal irregularities that stresses and weaken the material. By heating the steel at an intermediate temperature and cooling it, it allows to metal to rearrange itself in the atomic level so that it becomes more homogeneous, and stronger.

The same idea goes behind Quantum Annealing. The idea is to have lots of tiny little magnets called spins that possibly interacts with each other. Ordinarily, this configuration of spins may occupy a high energy state instead of the lowest energy state called the ground state. Classically, spins can get stuck in this higher energy state because they don’t have enough energy to get out of it. This is where the annealing portion comes in. To make sure that the spins occupy the ground state, one can heat up the system, giving the spins enough energy to reconfigure themselves such that they can enter a lower energy state, and then slowly decreasing the energy of the system. The quantum part of the process comes from the fact that the spins can occupy a superposition of states or be entangled with one another, in addition to being able to tunnel between energy states which are classically not allowed. The process ends when the spins occupy the lowest energy configuration, the ground state.

Now, this appears to have nothing to do with computation. Where is the answer to the computations? What is the problem being computed? Well, both the question and the answer turn out to be the ground state of the system.  It just so happens that when the number of spins get large, it becomes computationally very difficult to find out the ground state, but relatively simple to perform the Quantum Annealing process and just measure the ground state coming  out of it. So D-Wave’s system can only solve computational problems that can be modeled by a system of interacting spins, and where the desired answer is the ground state configuration of such a system of spins. This sounds a bit restrictive, and indeed it is, but that’s the device D-Wave is selling, and the only such device on the market, if it is not a complete fake of course.

One thing’s for sure though:  it isn’t going to give us the the Answer to the Ultimate Question of Life, the Universe, and Everything anytime soon.

Individuals usually lead fairly solitary lives; except on the occasions when they congregate for social activities, like, baseball or football. On these occasions they demonstrate a type of collective behavior which is motivated by their interests.
Do you know that we can find the same phenomenon in quantum physics? Distinguishable noninteracting particles work separately. However after interacting with another distinguishable particle, they often work together and demonstrate collective behavior. We call these phenomena composite particles. Do you know any of the well-known composite particles in physics? Ubiquitous examples include anyons, and cobosons. But there are a plethora of other examples which inspire a rapidly developing field of research.

Most quantum optics textbooks will spend a chapter or two on the theory of open quantum systems. The typical exposition consists of deriving the master equation in the Born-Markov approximation. This is a complicated integro-differential equation governing the evolution of the system state and further simplification requires the specification a concrete model Hamiltonian. The textbook will usually proceed to
give example Hamiltonians and derives the explicit form of the Born-Markov master equation. If you’ve had any exposure to open systems theory from a quantum optician’s perspective then you’ll know that all this is pretty standard. However, just because
it’s in a textbook (or “standard”) doesn’t mean it’s easy to understand. The Born-Markov approximation is rather abstract and explanations of it are often brief. Thus it leaves the reader a lot to swallow, or just simply stuck only after a couple of pages into the
chapter on open systems. Aside from getting the physical principles behind the approximation, the most difficult thing to swallow is that after making the approximation one is still left with a very complicated equation which is supposed to describe valid Markovian evolution for the system, and this is not at all obvious. Just seeing the Born-Markov master equation should make one lose heart that it really works. This is when knowing a bit of maths helps. We fight the abstract with the abstract. I’m referring to what is known as the Lindblad theorem which too often gets swept under the carpet in quantum optics texts. In brief this theorem states what the most general form of a Markovian master equation must look like, often called the Lindblad form*. Knowing this, the natural question is then whether the Born-Markov master equation leads to master equations in the Lindblad form? Lo and behold it does! So it appears then that one should just learn Lindblad’s theory and ditch Born and Markov since Lindblad’s result is much more general. Well, no, because Lindblad’s theorem is too general, mostly just formal mathematics. Lindblad’s theorem tells us what the correct answer has to look like but it doesn’t tell us how to get to the answer i.e. Lindblad’s theorem doesn’t provide a recipe for how to derive a Markovian master equation from model Hamiltonians (which is what physicists want). Learning Lindblad’s result does however settle any doubt that one had about the Born-Markov approximation when it was first encountered and this is great. Furthermore, it leads to a greater appreciation of the Born-Markov approximation for the reason that it actually points out the physics leading** to valid Markovian evolution. It is, in my opinion a remarkable fact that the Born-Markov master equation can be used to churn out a master equation in the Lindblad form. Similarly we can appreciate Lindblad’s work a lot more when we question the validity of the Born-Markov approximation. The moral of the story is that
Born-Markov and Lindblad complement each other and only by knowing both results do we get a complete picture of quantum Markovian dynamics. I believe therefore quantum
optics textbooks should present a more balanced approach to open systems theory by stating Lindblad’s theorem because this is when knowing a mathematical theorem actually allows one to appreciate the physics more.

*The Lindblad form should also be credited to Gorini, Kossakowski, and Sudarshan but I’ll refer to it as Lindblad just for convenience.

** I use the word “leads” because the master equation in the Born-Markov approximation alone does will not directly give us an equation in the Lindblad form.
In quantum optics one usually need to make a further approximation called the rotating-wave approximation to obtain the Lindblad form.

Every time we learn a new theory which is supposed to explain some physical phenomena we try to understand what is behind its axioms, why certain definitions were introduced and what is the physical meaning of the derived theorems. Let’s think about what it really means to understand and what we want to achieve from our study. Maybe it is fair enough just to be convinced about the significance of the theory without any deeper understanding.

Investigating a problem usually goes for searching for the logical reasoning between facts we find to be a cause of observed effects and using for this physical theories as a framework. To give an example, if we consider the planet movement we expect to take into account the gravitational interaction. Then we are able to foresee their future position but do we really get the underlying physics? The answer is yes; if we only wanted to describe another planetary system, we would succeed in doing so again. However, we cannot be satisfied, at this point we still do not know the interaction mechanism and cannot justify the formula for gravitational force.

With no doubt we should always examine the reasons for which the theory introduces its concepts. It is the best way to get to know all restrictions of the theory and to take a critical look at all made assumptions. On the other hand we will always face the wall of ideas and axioms, which not necessarily have to be intuitive.  The definition of kinetic energy was introduced by classical mechanics and became a part of the everyday language but still it is a purely theoretical notion. So it looks like we are condemned to play the game with known rules, we may even enjoy the game but will never find the explanation of all principles. We have to accept that there is a limit which we will never cross.

Quite often we just accept the validity of formulas and expressions proposed by other researchers. Maybe the reason of this attitude comes from the academic pressure. Having just a taste of the new theory we might be in the position to quickly identify some problems that could be solved within it. Things are fine as long as we remember to take the next step – going into the details of the theory. However, we can also say that using the elaborated model without considering all technical details does not have to indicate we are lazy. We may just want to get the new results based on what has been done by our predecessors.

It is very subjective when we can say, “OK, I’ve understood.” Maybe it is when we are ready to explain certain problems to others or we are just convinced about our knowledge and thanks to it we can predict the results of experiment. Most likely we will never succeed in our attempts to understand the whole nature. Thus the statement  “Still I don’t understand” takes the advantage as it drives our scientific progress.