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How procrastinating is sometimes a good thing

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Having worked with Quantum Mechanics for a couple of years, and getting comfortable with playing around with the rules and mathematics involved, a person may sometimes fall into the trap and say to himself: “Yeah, I think I understand Quantum Mechanics now.” But being a scientist, one is skeptical of everything, even the things you say to yourself, so he may go on further to ask “But do I really?” At times like these, I like to think of my favorite thought experiment – Wheeler’s Delayed Choice Experiment, following which I go back to thinking that quantum mechanics is this crazy thing, and all is right with the world again.

A good starting point to understand the delayed choice experiment is the (in)famous double slit experiment, which is basically present in any popular science book or textbook trying to explain what a weird thing quantum mechanics actually is. It has a simple enough setup: (i) you need a light source (ii) you shine said light source onto a thin plate with 2 slits (iii) you place a screen after that to see what comes out of the slits. Many people have tried this before, and what comes out of the slit is a wavelike pattern on the screen, with alternating light and dark spots. There is nothing inherently strange about this. We already know that light is a wave, and a wave will lead to a wavelike pattern on the screen on the other side of the double slit, big deal.

But wait. We also know that light can be thought of as being made up of tiny little nuggets called photons, so how does this explain the wavelike pattern on the screen? Well, light is typically made up of many, many photons right? Maybe all these photons passing through the slits are interacting amongst themselves and this results in an overall, collective wavelike pattern. This is a bit contrived, but still somewhat reasonable.

The problem comes about when the you have a light source that emits photons slowly, one at a time. If only one photon is passing through the slits at a time, then there isn’t anything it can possibly interact with. The logical end point of this is that if only one photon passes through the slits at a time, the wavelike pattern will disappear. However, countless experiments have been done to verify this, and all of them agree on one point: If you wait long enough and collect enough photons one at a time, the wavelike pattern persists. The only explanation for this is that the photon, although it is particulate in nature, nonetheless went through both slits at the same time and interacted with itself. Strange, but true. Funnily enough, if you observe which slit the photon actually passes through, and thus exclude the photon from ever passing through the other slit, then the wavelike pattern disappears. The photon behaves like a particle again.

The experiment could have just ended there – the conclusions to be drawn from it are strange enough. By this point, an experimenter could have relented and concluded “Okay, a photon is both a wave and a particle, strange, but at least I know that if the photon decides to go through both slits, it behaves like a wave coming out of it, and if it only went through a single slit, it behaves like a particle. I can accept that.” Nature, however, is relentless in trying to confound the experimentalist. This is where John Archibald Wheeler’s famous Delayed Choice Experiment comes in.

The delayed choice experiment is a simple variant of the classic double slit experiment, except performed by a lazy college student. The college student has one job: decide whether to put a screen and collect the overall pattern, or look at both of the slits to see which slit the photon went through (he may use a telescope to look at the slit, if that helps). Being ever the procrastinator, the college student decides what to do only at the last minute, long after the photon passes through the slits, and just before the photon reaches the college student. Now, since the photon had already passed through the slits, it must have decided to pass through either both slits, or a single slit, so you get a wavelike pattern for the former, and particle like behavior for the latter, regardless of what the student chooses to do. But no. For some reason, when the student puts up a screen, he gets a wave pattern, and if he looks at the slits to observe the photons, the wave pattern disappears. The conclusion? The you cannot even say that the photon went through a single slit or went through both slits. Somehow, the photon behaves like it went through both slits when the student puts up the screen, and behaves like it went through a single slit when the student observes which slit the photon went through, despite the fact that the photon already went through the slits long before the student even made his decision. The future influenced what happen in the past! Thus the experimenter was not even right to assume that the photons either went through both slits or a single slit. The most accurate answer we have is perhaps that the photon did both at the same time. And that’s just crazy if you think about it.

 

 

Second law of thermodynamics and heat engines

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The second law of thermodynamics is believed to be one of the most fundamental postulates of modern physics. Its most famous formulation is as the statement: the entropy of an isolated system never decreases; and it can be conveniently used as an excuse to never clean your room. This argument is ingeniously simple. Entropy of a macroscopic state is a measure of the number of possible corresponding microscopic configurations. In this case the macrostate is ‘a messy room’ and each microscopic configuration labels the positions of every object in that room such as the 10 books and a computer currently cohabiting your bed. Note that cleaning up a room involves ordering the objects in it. Since there are many possible configurations of these objects for which we say the room is messy but there may be only a small number of orderings for which we consider it clean; your efforts locally decrease the entropy of the room. However we know by the second law of thermodynamics that while we are doing the work to reordering the room we dissipate heat to the environment causing the universe to become more entropic globally; or in other words cleaning up your room contributes to the heat death of the universe and should only be undertaken at your own peril. Check out xkcd’s ideal room configuration.

This illustrates a perhaps more physical reinterpretation of the second law of thermodynamics; it is often restated as the impossibility of convert heat directly into work which ties into the following construction of a Heat engine. In this graphical representation we can see that not all the heat Q H transferred from the hot reservoir can be converted directly into useful work, only a fraction Q H – QC  can be accessed; the remainder must be dissipated to the cold reservoir thereby increasing the entropy of the cold reservoir. If we could completely convert Q H  directly into useful work then we could use this work to reduce the entropy (for example we could erase some information), at the same time we would have QC equal to zero so there would be no counterbalancing increase in the entropy of the cold reservoir. This would lead to a global entropy decrease thereby violating the second law. In our example of a messy room we realize that you are functioning as the hot reservoir and the universe (empty space has a temperature of about 3 Kelvin) is functioning as the cold reservoir. We can clearly see that it is sensible to define the amount of useful work which can be extracted from a system a concept called the Helmholtz free energy F. Making it possible to succinctly summarize everything we have said above by  the statement that  the change in Helmholtz free energy of the system can be at most equal to the average work done on a system: <W>  ≥  ΔF. The averaging over the work distribution in this statement arises because in finite systems we  can have quite large statistical fluctuations. This is an extremely powerful statement and a key tool in many active fields of research particularly in its relation to Szilard engines and generalized forms of the second law like Jarzynski equality which extends the second law of thermodynamics (relationship between free energy and work) to higher moments of the work distribution (the average work is the first moment).

 

Can we derive a non-contextual inequality from a simple game?

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Have you ever thought about whether we can derive an inequality, like the ones used in quantum information science, from any classic game? These inequalities help us identify the boundaries between classical and quantum behaviors.
Everybody knows the classic game, Rock-Paper-Scissors. It can be extended such as Rock-Paper-Scissors-Lizard-Spock, which was shown in “The Big Bang Theory”, one of the US TV network comedies. From this classic game, we can derive one of the non-contextual inequalities, i.e., KCBS inequality [PRL 101, 020403 (2008)].
Here is the rules:
1. Rock crushes lizard.
2. Lizard poisons Spock.
3. Spock smashes scissors.
4. Scissors cut paper.
5. Paper covers rock.
6. Rock breaks scissors.
7. Scissors decapitate lizard.
8. Lizard eats paper.
9. Paper disproves Spock.
10 Spock vaporizes rock.
The rules are referred to Wikipedia Rock-Paper-Scissors-Lizard-Spock. For two persons playing the game, we find that any fair player will win 2/5 of the time on average. If a player has only five different selection cards, then on average one person can defeat the other twice, i.e. 2/5 are multiplied by 5 selections. The scenario of winning the classical game gives us the same classical bound on a pentagram inequality in KCBS inequality. Coincidentally, the Rock-Paper-Scissors-Lizard-Spock game has a pentagram shape due to the relations (1-10) above.
What do you think? It might be some controversial; but it is interesting that we can try to derive another inequality from a game in our daily life :). To find out more about these contextual inequalities check out our latest review. Come out, come out, wherever it is!