Category Archives: The expert’s corner

You ask, we answer your questions related to the fascinating science at IQIM.

The TV Frontier

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Hello, my name is Tim Blasius and I am a physics graduate student at Caltech.  I recently appeared in a comedy bit on the TV show Conan, where I corrected Conan O’Brien’s physics.  I have been asked to share a few words describing this experience.

As with any story involving unbridled success, it begins as a tale of unnoticed, under-appreciated  and nearly unending hard work – I usually watch the show Conan during dinner with my fiancé.  Accordingly, I have seen many segments called “fan corrections” where Conan viewers submit YouTube videos explaining mistakes that they believe Conan has made on the show.  Some are nerdy and funny like the one where viewers pointed out that Conan used a red-tailed hawk call instead of a bald eagle call.  I was like a crocodile lurking in the water waiting for Conan to make a mistake in my expertise.  Then, like a woefully ignorant antelope sipping from the river Nile, Conan made a physics mistake when mocking Felix Baumgartner’s free fall, and I, being the bloodthirsty physics predator that I am, snapped the jaws of immutable truth around his naïve self.

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Quantum = Pink

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Need we say more?

What color do you imagine when you close your eyes and think “Quantum”? If you are to buy a case for your quantum computer, have you already picked your favorite color? (Okay, maybe it’s too early for that.)
Below I argue that the collective unconscious has already made the choice for you: it is going to be pink.

Excited?

Fear not. We will easily differentiate ourselves from warm and fluffy pink slippers. Our color is pink on black. Closer to purple, actually. We have good heritage: purple with white was the color of kings. But kings are no more, so let’s admit it: People think that “spooky” quantum phenomena have a purple glow around them. The disaster movie “Quantum Apocalypse” has a mysterious purple vortex approach Earth. Sci-fi now has “quantum cannons” shooting pink aura at the enemies, unleashing the chaos of uncertainty. You can’t fly your battlecruiser if you’re no longer certain you still have a battlecruiser.
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Fermat’s Lost Theorem

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Pierre Fermat, known for his Last Theorem and for rarely proving any of his claims.


Last week, I posted a series of increasingly difficult challenges in the post One line proof. Today, I would like to spend some time with the second challenge:

Fermat’s Lost Theorem: Show that (x+y)^n = x^m+y^m has only one solution for integers x > y > 0 and m,n > 1.

The truth is, I don’t know how to solve this problem myself. But, I think that we can figure it out together. Below, I will give the solution to the simpler case of y=1 and x > 1. I expect that many of you know the following variant of the problem:

Fermat’s Last Theorem: Show that z^m = x^m + y^m has no solutions for integers z > x > y > 0 and m > 2.

The above theorem was one of the most important unsolved problems in mathematics, until Andrew Wiles presented his proof to the public at a conference in Cambridge in 1993. Then someone pointed out a serious flaw in his proof and the extreme high Wiles was feeling turned into a dark abyss of despair. But being awesome implies that you pick yourself up and run full force towards the wall as if you didn’t get floored the last time you tried breaking through. And so Andy went back to his office and, with a little help from his friend (and former student) Richard Taylor, he fixed the flaw and published the massive proof in the most prestigious journal of mathematics, the Annals of Mathematics, in 1995.
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One line proof

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It is not often that we come across a problem whose solution can fit in the margins of a notebook. In fact, many of the problems I have worked on in the field of quantum many-body physics require proofs that often exceed 30 pages. And that is without taking into account the several references included as sources for results used as “elementary” tools in the proof (referees love these papers…) So it is natural to think that a proof is a proof, so long as it is correct, and once confirmed by the academic community it is time to move on to something new.

But… sometimes things get interesting. A 30-page proof collapses to a 3-page proof when a different point of view is adopted (see the famous Prime Number Theorem). Below, you will find two problems that may, or may not have a “one-line” proof. The challenge is for you to find the shortest, most elegant proof for each problem:

A unit triangle: A triangle with sides a\le b \le c has area 1. Show that b^2 \ge 2.

Fermat’s Lost Theorem: Show that (x+y)^n = x^m+y^m has only one solution with positive integers x > y > 0 and m,n > 1.

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Individual quantum systems

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When I went to school in the 20th century, “quantum measurements” in the laboratory were typically performed on ensembles of similarly prepared systems. In the 21st century, it is becoming increasingly routine to perform quantum measurements on single atoms, photons, electrons, or phonons. The 2012 Nobel Prize in Physics recognizes two of the heros who led these revolutionary advances, Serge Haroche and Dave Wineland. Good summaries of their outstanding achievements can be found at the Nobel Prize site, and at Physics Today.

Serge Haroche developed cavity quantum electrodynamics in the microwave regime. Among other impressive accomplishments, his group has performed “nondemolition” measurements of the number of photons stored in a cavity (that is, the photons can be counted without any of the photons being absorbed). The measurement is done by preparing a Rubidium atom in a superposition of two quantum states. As the Rb atom traverses the cavity, the energy splitting of these two states is slightly perturbed by the cavity’s quantized electromagnetic field, resulting in a detectable phase shift that depends on the number of photons present. (Caltech’s Jeff Kimble, the Director of IQIM, has pioneered the development of analogous capabilities for optical photons.)
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No calculators allowed

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During Friday evenings at Caltech, the Institute for Quantum Information and Matter runs a seminar series focusing on research talks geared towards a diverse audience whose common background is quantum mechanics (just that, nothing else…) But, the fun doesn’t end after the 45 minute talks are over. The seminar is followed by a mingling session on the brand new patio of East Bridge, the building where Richard Feynman delivered his famous lectures.

Grigori Perelman. Not a Russian middle-schooler anymore, but still pretty smart.

Last Friday, Aleksander Kubica (pronounced Koobitsa), a Polish graduate student in the Theory group of IQIM and a first prize winner of the 2009 European Union Contest for Young Scientists, decided to test our ingenuity by giving us a problem that Russian middle-schoolers are expected to solve. If you haven’t met any Russian middle-schoolers, well, let’s just say that you should think carefully before accepting a challenge that is geared towards their level of mathematical maturity. What was the challenge?

Problem 1: Show that: \frac{1}{2}\cdot \frac{3}{4}\cdot \frac{5}{6}\cdot \frac{7}{8}\cdot \frac{9}{10}\cdots \frac{99}{100} < \frac{1}{10}.
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Redemption: Part II

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Last week, a journey began to find the solution to a problem I could not solve as a seventeen year-old boy. That problem became an obsession of mine during the last days of the International Math Olympiad of 1997, the days when I also met the first girl I ever kissed. At the time, I did not have the heart to tell the girl that I had traveled across the Atlantic to compete with the best and brightest and had come up short. I told her that I had solved the problem, but that the page with my answer had been lost. I told my parents the same thing and to everyone at school who would ask me why I did not return with a medal from the Math Olympics. The lie became so powerful that I did not look at that problem again until now. So, you may be wondering why a blog about Quantum Information Science at Caltech includes posts on problems from Math Olympiads. And why I would open the book on the page with that one problem after fifteen years… Continue reading →

Redemption: Part I

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Back in my last post, I closed with a problem (The Redeemer) from the International Math Olympiad of 1997. Unlike the problem I posted from 1981 (Elementary, my dear Watson), I was actually walking by the time I met The Redeemer. In fact, I was in Mar del Plata, a beautiful seaside beach resort in Argentina, participating in the 38th International Math Olympiad. The Redeemer was Problem 5. I remember clearly the moment I saw it. It looked so simple, so innocent: Find all pairs of positive integers a,b \ge 1, such that a^{(b^2)} = b^a.

I had to solve it. But little did I know…

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The allure of elegance

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On one of my many short-lived attempts to study for the International Physics Olympiad, I picked up a copy of a well-known textbook by Halliday and Resnick. My high school didn’t offer a course in electricity and magnetism, so I figured I would have to teach myself about the topic if I wanted to solve problems competitively.

I only managed to solve about 3 problems from the book before a friend called me up asking if I wanted to play Frisbee. After that, I don’t think I ever looked at the text again, although for some reason I did bring it to college – maybe to make sure I had enough things gathering dust on my dorm room shelf. That’s not to say it’s a bad book! In fact, I learned quite a bit from the three problems I solved. Here’s one of them:

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Three Questions: Zach Korth, LIGO

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Who are you?

My name is Zach Korth. I’m a graduate student here at Caltech, working in Prof. Rana Adhikari’s group on experimental gravitational wave physics. The bulk of my time goes to developing and testing technology that will be installed at the detectors of the Laser Interferometer Gravitational-wave Observatory (LIGO) in Livingston, LA, and Hanford, WA. Run jointly by Caltech and MIT, LIGO is poised to make the first ever direct detection of gravitational waves, ripples of space-time itself propagating across the universe at the speed of light, carrying with them information about the most distant and poorly understood astrophysical phenomena thought to exist.

At the LIGO Hanford control room.


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