DJ Strouse

A few weeks ago, I began a project with USC Professor Daniel Lidar to improve the efficiency of a nifty little procedure called “interaction-free measurement.” Here’s the picture:

A terrorist tells you he has planted a super-sensitive bomb in one hallway of a building, but you don’t know which one. The bomb will detonate if a single photon interacts with it (presumably the building’s hallways are naturally pitch dark to begin with). How do you detect where the bomb is?

Sounds like you’re screwed, right? Not if you know a bit of quantum mechanics. In 1993, Avshalom Elitzur and Lev Vaidman showed how to exploit the interference effects in sending a photon through an interferometer to detect the bomb without interacting with it. Don’t get too excited yet though. Unfortunately, their scheme also detonated twice as many bombs as it detected interaction-free.

A few years later, Kwiat and friends figured out that if you cycle that photon through the interferometer many times, you can reduce the ratio of detonations to interaction-free measurements arbitrarily low, using another nifty trick known as the quantum Zeno effect.

More recently, a few physicists at Lousiana State University pointed out that any realistic super-sensitive bomb detection (*cough*) would take place in the presence of noise and that Kwait and friends’ little game would fail. But don’t worry; we’re still safe from evil-doers. They showed that with a few other tricks, we can still reduce our detonation to interaction-free measurement ratio to as tiny and safe a value as we please.

Just one more caveat – it may take a while. Their scheme might require many, many unsuccessful trials that neither detonate the bomb nor determine whether it is present before ever getting a certain result. So if this bomb has a timer on it, we still might be in trouble.

What Professor Lidar and I are trying to do is make the bomb-testing procedure even more efficient by changing pieces of the interferometer every time the photon cycles. By optimizing over a couple different parameters (including the phase rotation angle and possibly the photon loss in arm of the photon that possibly includes the bomb), we hope to reduce the number of trials expected in order to stealthily detect the bomb.

I spent a few hours today trying to represent the components that make up the efficiency in a simpler form amenable to further analysis and optimization. By massaging this formula into something simpler, Professor Lidar and I hope we can (a) better understand how to optimize it and (b) prove bounds that show that our optimization is actually optimal (or at least close).

Optimization is a new game to me so, while I still don’t have clear intuitions about the most useful representations and approximations, I’m excited to learn some new tricks. Not only is optimization useful for many problems in physics, mathematics, and science in general, its one of those subjects which is irresistibly fun to haphazardly apply to every day life. (As Stanford’s Stephen Boyd once remarked, “Every student of optimization will at one point come to the seemingly profound but misguided realization, ‘Oh my god… everything in life is an optimization problem!‘”). I’ll also be taking a course in convex optimization this fall which should be both useful and a lot of fun.

I’ve learned at least two important things so far while working on this project. One, reproducing the figures in a paper is a great way to test your understanding of the ideas. Scientific papers are notoriously terse and many details are left out, so filling the gaps and piecing together the story embedded in the figures is a nice little challenge. Two, beginning with the simplest cases is often essential, even when you know you’re interested in the more complicated ones. It’s tempting to try and tackle the exact problem you’re interested in right away, but beginning with simpler cases both allows you to better understand the problem itself and practice with methods of analysis that will likely be useful for the complicated case too. Think of them like scientific push-ups.

While things don’t look to be simplifying much in my current calculations, I’ll give them another stab on Thursday.