Category Archives: The expert’s corner

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

A poll on the foundations of quantum theory

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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.
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Science books for kids matter (or used to)

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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.

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One, two, three, four, five…

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As this year comes to a close, people around the world will be counting down the last few seconds of 2012. But, how come we never count up the first few seconds of the new year? What is it about the last few seconds of the year that makes them so special? Maybe it has to do with surviving a Mayan apocalypse, but that was just this year. I guess it always comes down to letting go of the good and of the difficult moments in our past. The champagne helps. Still, I would like to take a moment to pay tribute to the first few seconds of 2013 (the future) with a simple problem and a twist…

back-to-the-future

The question is simple enough:
What is the longest sequence of consecutive numbers, such that each element of the sequence is a power of a prime number?

You will arrive at the answer soon after asking yourself the question: How often is a number divisible by both 2 and 3? The answer is every 6 numbers (since the number in question has to be divisible by 2\cdot3). So, the best one can do is to count the numbers from one to five. That is, if you count 1 as the power of a prime number, then the first five numbers have the incredible property of being the only such sequence of numbers (right?). [Note: Recalling that 1 is not a prime number (see Redemption: Part I, for a hint), we will allow ourselves to use 0 as a valid power to which we may raise a prime number, thus getting 2^0 = 1, which completes the argument.]

Now, here comes the twist…

Challenge: Is there a sequence of 10 consecutive numbers such that none of them is a power of a prime number?

To get the ball rolling, here are the first such sequences with one, two and three elements, respectively: [6], [14,15], [20,21,22].

Super Challenge: Can you find a sequence of 2013 consecutive numbers, such that none of them is a power of a prime number?

Impossible Challenge: Can you solve the first challenge, giving the sequence containing the smallest numbers satisfying the conditions of the problem?

Good luck and enjoy the rest of 2012! Who knows, maybe some genius will solve the above challenges before 2013 rolls around…

Introduction to Quantum Information

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First slide, viewed on my laptop.

First slide, viewed on my laptop.

I’m lazy. The only reason I ever do anything is that sometimes in a weak moment I agree to do something, and after that I don’t have the nerve to back out. And that’s how I happened to give the introductory lectures leading off the 12th Canadian Summer School on Quantum Information last June.

The video of the lectures recently became available on YouTube in two one-hour segments, which is my reason for posting about them now:

Here are the slides I used. The school is pitched at beginning graduate students who have a solid background in quantum mechanics but may not be very familiar with quantum information concepts.

Andrew Childs, who knows my character flaws well, invited me to lecture at the school nearly a year in advance. Undaunted by my silence, he kept resending the invitation at regular intervals to improve his chances of catching me on a weak day. Sure enough, feeling a twinge of guilt over blowing off David Poulin when he made the same request the year before, and with a haunting sense that I had refused to do something Andrew had asked me to do on an earlier occasion (though I can’t recall what), one day in September I said yes, feeling the inevitable stab of regret just seconds after pushing the Send button. I consoled myself with the thought that this could be a Valuable Service to the Community.

Actually, it was fun to think about what to include in my lectures. The job was easier because I knew that the other lecturers who would follow me, all of them excellent, would be able to dig more deeply into some of the topics I would introduce. I decided that my first responsibility should be to convey what makes the topic important and exciting, without getting too bogged down in technicalities which were likely to be addressed later in the school. That meant emphasizing the essence of what makes quantum information different from ordinary “classical” information, and expounding on the theme that classical systems cannot in general simulate quantum systems efficiently.

The conditions under which I delivered the lectures were not quite ideal. Preparing PowerPoint slides is incredibly time consuming, and I believe in the principle that such a task can fill however much time is allotted for it. Therefore, as a matter of policy, I try to delay starting on the slides until the last moment, which has sometimes gotten me into hot water. In this case it meant working on the slides during the flight from LA to Toronto, in the car from Toronto to Waterloo, and then for a few more hours in my hotel room until I went to bed about midnight, with my alarm set for 6 am so I could finish my preparations in the morning.

It seemed like a good plan. But around 2 am I was awakened by an incredibly loud pounding, which sounded like a heavy mallet hammering on the ceiling below me. As I discovered when I complained to the front desk, this was literally true — they were repairing the air-conditioning ducts in the restaurant underneath my room. I was told that the hotel could not do anything about the noise, because the restaurant is under different ownership. I went back to bed, but lost patience around 3:30 am and demanded a different room, on the other side of the hotel. I was settled in my (perfectly quiet) new room by 4 am, but I was too keyed up to sleep, and read a book on my iPad until it was 6 am and time to get up.

I worked in my room as late as I could, then grabbed a taxi, showing the driver a map with the location of the summer school marked on it. Soon after he dropped me off, I discovered I was on the wrong side of the University of Waterloo campus, about a 20 minute walk from where I was supposed to be. It was about 8:15, and the school was to begin at 8:30, so I started jogging, though not, as it turned out, in the right direction. After twice asking passersby for help, I got to the lecture hall just in time, my heart pounding and my shirt soaked with sweat. Not in the best of moods, I barked at Andrew that I needed coffee, which he dutifully fetched for me.

Though my head was pounding and my legs felt rubbery, adrenalin kicked in as I started lecturing. I felt like I was performing in a lower gear than usual, but I wasn’t sure whether the audience could tell.

And as often happens when I reluctantly agree to do something, when it was all over I was glad I had done it.

Is Alice burning? The black hole firewall controversy

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Quantum correlations are monogamous. Bob can be highly entangled with Alice or with Carrie, but not both.

Quantum correlations are monogamous. Bob can be highly entangled with Alice or with Carrie, but not both.

Back in the early 1990s, I was very interested in the quantum physics of black holes and devoted much of my research effort to thinking about how black holes process quantum information. That effort may have prepared me to appreciate Peter Shor’s spectacular breakthrough — the discovery of a quantum algorithm for factoring intergers efficiently. I told the story here of how I secretly struggled to understand Shor’s algorithm while attending a workshop on black holes in 1994.

Since the mid-1990s, quantum information has been the main focus of my research. I hope that some of the work I’ve done can help to hasten the onset of a new era in which quantum computers are used routinely to perform super-classical tasks. But I have always had another motivation for working on quantum information science — a conviction that insights gained by thinking about quantum computation can illuminate deep issues in other areas of physics, especially quantum condensed matter and quantum gravity. In recent years quantum information concepts have begun penetrating into other fields, and I expect that trend to continue.
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Showtime for Sophomores

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Exciting a square Chladni plate with a violin bow.

One of the many unique features of Caltech is our core curriculum. All of our undergraduates are required to take five terms of physics and five terms of math (all three terms freshman year and the fall and winter terms sophomore year) — though this will change for the class entering in the fall of 2013.

Each fall, about 170 sophomores take Physics 2a, a course on vibrations, waves, and quantum mechanics, while the remaining 60 or so sophomores take Physics 12a, a souped up course covering similar material at a level more appropriate for physics concentrators.

This term I am teaching Physics 2a. While 170 students is a lot more than in most courses I teach at Caltech, the workload is manageable, in part because I share the lecturing duties with another professor, and in part because we have a staff of capable and hard working Teaching Assistants who handle recitation sections and grade the homework and quizzes.

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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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