Unsolvable

mar-de-plata1

Why go swimming, if you can do math instead inside a room with no windows?

Back in 1997, during my visit to beautiful Mar del Plata in Argentina, I was asked to solve a math problem that I soon realized was close to being unsolvable. The setting was the Banquet for the 38th International Math Olympiad. I was 17 and there was delicious, free food in front of me, so it was pretty impossible to get my attention. Still, the coach of the Romanian team decided to drop by the Greek table to challenge us with the following problem:

Infinite Power: If x^{x^{x^{x^{\dots }}}} = 2, find x.

I was pretty hungry and a bit annoyed at the interruption (it is rude to eat and do math at the same time!), so I looked at the problem for a moment and then challenged him back with this one:

Infiniter Power: If x^{x^{x^{x^{\dots }}}} = 4, find x.

After a few moments, he looked at me with a puzzled look. He knew I had solved his problem within a few seconds, he was annoyed that I had somehow done it in my head and he was even more annoyed that I had challenged him back with a problem that confounded him.

Your mission, if you choose to accept it, is to figure out why the coach of the Romanian IMO team left our table alone for the rest of the competition.

Good luck!

PS: The rest of the Romanian team (the kids) were really cool. In fact, a few years later, I would hang out with a bunch of them at MIT’s Department of Mathematics, or as my older brother put it, The Asylum.

Science Magazine’s Breakthrough of 2012

A few nights ago, I attended Dr. Harvey B. Newman’s public lecture at Caltech titled: “Physics at the Large Hadron Collider: A New Window on Matter, Spacetime and the Universe.” The weekly quantum information group meeting finished early so that we could attend the lecture (Dr. Preskill’s group meeting lasted slightly longer than two hours: record brevity during the seven months that I’ve been a member!) We weren’t alone in deciding to attend this lecture. Seating on the ground floor of Beckman Auditorium was filled, so there were at least 800 people in attendance. Judging by the age of the audience, and from a few comments that I overheard, I estimate that a majority of the audience was unaffiliated with Caltech. Anyways, Dr. Newman’s inspiring lecture reminded me how lucky I am to be a graduate student at Caltech and it also clarified misconceptions surrounding the Large Hadron Collider (LHC), and the discovery of the Higgs, in particular.

Before mentioning some of the highlights of Dr. Newman’s lecture, I want to describe the atmosphere in the room leading up to the talk. A few minutes before the lecture began, I overheard a conversation between three women. It came up that one of the ladies is a Caltech physics graduate student. When I glanced over my shoulder, I recognized that the girl, Emily, is a friend of mine. She was talking to a mother and her high school-aged daughter who loves physics. It’s hard to describe the admiration that oozed from the mother’s face as she spoke with Emily–it was as if Emily provided a window into a future where her daughter’s dreams had come true. It brought back memories, from when I was in the high schooler’s position. As a scientifically-minded child growing up in Southern California, I dreamed of studying at Caltech, but it seemed like an impossible goal. I empathized with the high schooler and also with her mother, who reminded me of my own mom. Mom’s have a hopeless job: they’re genetically programmed to want the best for their children, but they oftentimes don’t have the means to make these dreams a reality. Especially when the child’s dream is to become a scientist. It’s a rare parent who understands the textbooks that an aspiring scientist consumes themselves with, and an even rarer parent, who can give their child an advantage when they enter the crapshoot that is undergraduate admissions. The angst of the conversation reminded me that I’m one of the lucky few whose childhood dreams have come true–it’s an opportunity that I don’t want to squander.

The conversation between two elderly men sitting next to me also brought back uncomfortable memories. They were trying to prove their intelligence to each other through an endless proceeding of anecdotes and physics observations. I empathized with them as well. Being at a place like Caltech is intimidating. As an outsider, you don’t have explicit credentials signaling that you belong, so you walk on eggshells, trying to prove how smart you are. I’ve seen this countless times, such as when I give tours to high schoolers, but it’s especially pronounced amongst incoming graduate students. However, it quickly fades as they become comfortable with their position. But to outsiders, every time they re-enter a hallowed place, their insecurities flood back. I know this because I was guilty of this! I spoke with the gentlemen for a while and they were incredibly nice, but smart as they were, they were momentarily insecure. Putting on my ambassador hat for a moment, if there are any ‘outsiders’ reading this blog, I want to say that I, for one, am glad that you attend events like this.
Continue reading

Topological insulator trio recognized by 2013 Physics Frontiers Prize

I was excited to hear that Charlie Kane, Laurens Molenkamp and Shoucheng Zhang were among the recipients of the 2013 Physics Frontiers Prize. Their seminal works both theoretically predicting and experimentally discovering the topological insulator have profoundly influenced the direction of condensed matter physics over the past few years and have shaped my own research agenda at Caltech.

I first learned about the topological insulator from a talk Charlie Kane delivered at the 2006 American Physical Society March Meeting. It was around this time that graphene was exploding onto the scene following Andre Geim and Konstantin Novoselov’s demonstration that single sheets of it could be peeled from graphite using Scotch tape. Inspired by the highly unconventional charge transport properties that were being measured from graphene, Charlie had begun to think about whether its spin transport properties might also yield surprises. The huge surprise, as Charlie would reveal in his talk, was that graphene could theoretically exhibit a quantum spin Hall effect in which spin-polarized charge carriers flow without dissipation along the edges of an electrically insulating material. Such quantum spin Hall insulators (later renamed the 2D topological insulator), as Charlie and his colleague Eugene Mele proved, are a phase of matter distinct from ordinary electrical insulators by virtue of a quantum entanglement of its electrons.
Continue reading

A poll on the foundations of quantum theory

Erwin Schrödinger. Discussions of quantum foundations often seem to involve this fellow's much abused cat.

Erwin Schrödinger. Discussions of quantum foundations often seem to involve his much abused cat.

The group of physicists seriously engaged in studies of the “foundations” or “interpretation” of quantum theory is a small sliver of the broader physics community (perhaps a few hundred scientists among tens of thousands). Yet in my experience most scientists doing research in other areas of physics enjoy discussing foundational questions over coffee or beer.

The central question concerns quantum measurement. As often expressed, the axioms of quantum mechanics (see Sec. 2.1 of my notes here) distinguish two different ways for a quantum state to change. When the system is not being measured its state vector rotates continuously, as described by the Schrödinger equation. But when the system is measured its state “collapses” discontinuously. The Measurement Problem (or at least one version of it) is the challenge to explain why the mathematical description of measurement is different from the description of other physical processes.

My own views on such questions are rather unsophisticated and perhaps a bit muddled:

1) I know no good reason to disbelieve that all physical processes, including measurements, can be described by the Schrödinger equation.

2) But to describe measurement this way, we must include the observer as part of the evolving quantum system.

3) This formalism does not provide us observers with deterministic predictions for the outcomes of the measurements we perform. Therefore, we are forced to use probability theory to describe these outcomes.

4) Once we accept this role for probability (admittedly a big step), then the Born rule (the probability is proportional to the modulus squared of the wave function) follows from simple and elegant symmetry arguments. (These are described for example by Zurek – see also my class notes here. As a technical aside, what is special about the L2 norm is its rotational invariance, implying that the probability measure picks out no preferred basis in the Hilbert space.)

5) The “classical” world arises due to decoherence, that is, pervasive entanglement of an observed quantum system with its unobserved environment. Decoherence picks out a preferred basis in the Hilbert space, and this choice of basis is determined by properties of the Hamiltonian, in particular its spatial locality.
Continue reading

Science books for kids matter (or used to)

The elementary school I attended hosted an annual book fair, and every year I went with my mother to browse. I would check out the sports books first, to see whether there were any books about baseball I had not already read (typically, no). There was also a small table of science books, and in 1962 when I was in the 4th grade, one of them caught my eye: a lavishly illustrated oversized “Deluxe Golden Book” entitled The World of Science.

My copy of The World of Science by Jane Werner Watson, purchased in 1962 when I was in the 4th grade.

My copy of The World of Science by Jane Werner Watson, purchased in 1962 when I was in the 4th grade.

As I started leafing through it, I noticed one of the cutest girls in my class regarding me with what I interpreted as interest. Right then I resolved to buy the book, or more accurately, to persuade my mother to buy it, as the price tag was pretty steep. Impressing girls is a great motivator.

The title page.

The title page.

Continue reading

Ignacio Cirac and Peter Zoller get what they deserve

Ignacio Cirac, Dave Wineland, and Peter Zoller receiving the 2010 Franklin medal.

Ignacio Cirac, Dave Wineland, and Peter Zoller receiving the 2010 Franklin medal.

A good thing about a blog is that when my friends win prizes I have the opportunity to say nice things about them. This seems to be happening a lot lately (Kitaev, Wineland, Kimble, Hawking, Polchinski, …).

Today’s very exciting news is that Ignacio Cirac and Peter Zoller have won the 2013 Wolf Prize in Physics “for groundbreaking theoretical contributions to quantum information processing, quantum optics, and the physics of quantum gases.”
Continue reading