Monthly Archives: March 2009

Hardy’s Paradox, or The Economist is Dismal at Science

Today let’s talk about Hardy’s Paradox, since I’m guessing you haven’t yet had your daily dose of quantum mechanics conversations.

I got interested in this topic after Jeff pointed me at The Economist’s writeup of a recent confirmation of a puzzling aspect of quantum mechanics.

What the several researchers found was that there were more photons in some places than there should have been and fewer in others. The stunning result, though, was that in some places the number of photons was actually less than zero. Fewer than zero particles being present usually means that you have antiparticles instead. But there is no such thing as an antiphoton (photons are their own antiparticles, and are pure energy in any case), so that cannot apply here.

That can’t be right, I thought, so I checked out the original paper. The paper is strikingly devoid of negative photons, though I understand why the Economist tripped up trying to explain negative weak values. I’m not sure I can really explain them, either. But I can try to explain Hardy’s paradox, how physicists can now measure things without disturbing the system, and why you can’t always figure out the past by looking at the present.

To start, let’s talk about a quantum basketball fan — a Duke fan, naturally. It’s halftime at the Duke-UNC game, so he goes to buy a drink. There are two ways he can get to the refreshment stands, and two stands to choose from, one selling Coke, the other Pepsi. (For the optical physicists among you — hi, Mike and Michael! — I’m setting up a Mach-Zender interferometer.) If he takes the red hallway, he’ll end up at the Coke stand. If he takes the blue one, he’ll end up at the Pepsi stand. Since he’s a quantum basketball fan, he is both a particle and a wave, and can take both hallways at once. What’re his odds of ending up at the Pepsi stand instead of the Coke one? If he were to pick a hallway purely at random, he’s got a 50/50 chance.

A Duke fan going to buy either a Coke or a Pepsi.

That doesn’t take interference into account, though. See, waves can interfere with each other. You can see that with light. If you set things up right, you can have light interfere constructively at a given spot, making a bright spot, or you can have it interfere destructively, so there is no light at that spot. The same thing happens with any wave, and can in fact happen to our quantum basketball fan. We’ll set the length of the hallways so that, through the magic of interference, he doesn’t have a 50/50 chance of ending up at either stand. He only ends up at the one selling Coke, and never at the one selling Pepsi. He only goes through the red hallway and not the blue one. The only way he can end up at the Pepsi stand is if there’s something else in the red hallway that affects him, like another fan.

Waves interfering with each otherThe funny thing about quantum interference, though, is that, if we watch the guy to see which hallway he takes, he’ll take one or the other, and can end up at either refreshment stand. By measuring which path he takes, we keep him from acting as a wave.

Back to our basketball game. Since this is halftime, there’s also a UNC fan who wants to go buy a Coke. He’s got his own set of hallways and own set of refreshment stands. And his hallways are set up so he always ends up at the Coke stand instead of the Pepsi stand. But whatever genius built the stadium had the Duke fan’s hallways meet up with the UNC fan’s hallways at one point. That’s bad news: if the Duke and UNC fans meet, they’ll get into a fist fight and knock each other out, and neither will get to the refreshment stands.

Overlapping hallways involving the Duke and UNC fans

Here’s where I’m going to blow your mind. According to quantum mechanics, there’s now a chance that both fans will end up at the Pepsi stands. But wait! The only way that the Duke fan can end up at the Pepsi stand is if the UNC fan got in his way and messed up his self-interference, and that only happens if both guys go through the red hallways that meet. But if the Duke and UNC fans meet, then neither can get to the refreshment stands because they’ll fight! Since the Duke fan got to a refreshment stand, he can’t have met the UNC fan. But since he got to the Pepsi stand, he must have met the UNC fan! But since he met the UNC fan, he can’t have gotten to the Pepsi stand because of the fight!

That’s Hardy’s paradox. He used electrons and positrons in overlapping Mach-Zender interferometers, but my version is more likely to get me sponsors.

For a while, people have tap-danced around Hardy’s paradox. The problem is that you’re trying to perform retrodiction, which is like a prediction about the past. You’d think retrodiction would be easier than prediction, but in this case it’s not. We don’t know exactly what’s going on inside those hallways, and yet we’re trying to say something about the path the fans took by seeing where they ended up. Traditionally physicists have said, “There isn’t a good physical interpretation for things we don’t measure.” If we try to see which hallway the fans took, we’ll destroy the very effect we’re trying to measure, because they’ll stop acting like waves.

What if we could measure the system without disturbing it? Is there a way to see what hallway a fan took without actually looking? The absolute answer is “no”. Any measurement we make disturbs the system. But if we’re willing to make an imprecise measurement, we can keep from disturbing the system too much. In interaction-free measurement, you measure so imprecisely that you don’t affect the system. It’s like saying that if we squint and don’t get a very clear view of the hallways, we can kind of see which hallway each fan went down. These weak measurements are noisy as all get-out, but by running the experiment over and over with lots and lots of Duke and UNC fans and averaging the results, we can get a clear picture of what’s going on.

Back in 2001, Aharonov et. al. suggested that you could see Hardy’s paradox in action through weak measurements (as published in Phys. Lett. A). Let’s go back to the basketball fans. Label the hallways in terms of whether they overlap (i.e. they meet) or they don’t: the red hallways overlap, while the blue ones are non-overlapping.

Overlapping and non-overlapping hallways

Aharonov and his colleagues worked through the math to answer questions like “which way does the Duke fan go?” and “which way does the UNC fan go if the Duke fan goes through the hallways that overlap?” There are actually two sets of questions: what does each fan do individually, and what do both fans do at the same time? Here’s the probabilities for the individual fans:

Situation Probability
Duke fan goes through overlapping (red) hallway 1
Duke fan goes through non-overlapping (blue) hallway 0
UNC fan goes through overlapping (red) hallway 1
UNC fan goes through non-overlapping (blue) hallway 0

That’s what we’d expect. Each fan has no chance of going through the blue hallways and is guaranteed to go through the red ones. But what if we consider what both the fans do at the same time?

Situation Probability
Duke and UNC fans go through overlapping (red) hallways 0
Duke fan goes through overlapping (red) hallway; UNC through non-overlapping (blue) 1
Duke fan goes through non-overlapping (blue) hallway; UNC through overlapping (red) 1
Duke and UNC fans go through non-overlapping (blue) hallway -1

Both fans are guaranteed not to go through the red hallways, because if they did, they’d meet and get in a fight. There’s Hardy’s paradox! But even weirder, there’s a -1 probability that they both go through the blue hallways. That’s clearly nonsense: there’s no such thing as a negative probability. Or if you think about it in terms of fans, quantum mechanics says that there’s -1 pairs of fans in those hallways!

Before, I might have said, “That’s no problem, because you can’t measure both the individual fans’ path and their path together at the same time.” But with weak measurements you can! And in fact, Lundeen and Steinberg (as published in Phys. Rev. Letters in January) and Yokota, Yamamoto, Koashi, and Imoto did. They used photons and their polarization instead of electrons or basketball fans, but they experimentally confirmed the above results.

It turns out weak-valued probabilities don’t have to be positive definite, but what does that mean? In their paper, Lundeen and Steinberg say

Recall that the joint values are extracted by studying the polarization rotation of both photons in conicidence…. As in all weak measurement experiments, a negative weak value implies that the shift of a physical “pointer” (in this case, photon polarization) has the opposite sign from the one expected from the measurement interaction itself.

In their experiment, they were measuring photon polarizations. They saw that, for the -1 case, it was as if the photons’ polarizations had shifted in the opposite direction than they should have. So it’s not true, as The Economist said, that the number of photons was ever “less than zero” at any location. You can’t hang such classical concepts on this quantum mechanical effect. Now that I’ve read the papers I can see what The Economist writer was trying to say, and I’m not sure I could have done better given space constraints — look how long I’ve talked about it here!

I still don’t have a good understanding of what a -1 weak probability really means, but it’s a surprising and neat resolution to the paradox. The -1 makes the math square up. Everything is self-consistent, even if it is weird. Hardy’s paradox isn’t really a paradox from the view of quantum mechanics.

The Dark Secret of Wall-E

In lieu of actual original content from me, watch Jay Smooth lay out a frighteningly plausible theory about Wall-E and why all the humans in the movie were white middle-class ones.

Jay is the man behind the awesome “How to Tell People They Sound Racist” video from last year. In fact, let me throw that one in as well. Two videos make a blog post, right?

Cross-Stitch Purse & Other Craft Goodness

A few years ago I saw a cross-stitch magazine with a plastic box purse on the cover. The sides were cross-stitch flowers, stripes, and funky circles. I bought the magazine thinking I would do the project immediately. Then I saw how much the purses cost. They are awesomely designed purses from Z Becky Brown but the price tag is not for the faint of heart. I shelved the magazine.

Last summer I visited a locally-owned craft store and there among the closeout specials was one single Z Becky Brown purse. And it was in my price range! I bought it and ran out of the store before the clerk knew that they were losing money on my transaction. Needless to say, I found the pattern and began work. I’ve worked on and off on the stitching since August. I finished it today. Tra-la-la! Here it is:

Hibiscus Purse

Back of the Hibiscus Purse

I also finished another short project today. My contribution to The Toy Society:

Blue Bird of Happiness

I’ve actually had the bird done for a while but hadn’t had a plan for him. Today I made up the tag and the note explaining what The Society does to go with him. I’m going to be in downtown Huntsville on Wednesday so will leave him someplace where he hopefully will be found by a child in need of a toy.

So I’m sure you’re asking yourself, “What will she work on next?” I’ll be starting a portrait of Erwin Schrödinger from Greta at Larripin Labs.


Greta has created some nifty portraits of famous physicists. I asked Stephen if he’d be interested in having one and he picked this one. I’m going to be making some color alterations so stay tuned to see if I can make it all come out right.

I’ll also be working on a Mario Mushroom from the Friendly Flamingo for Eli. It’s tiny but oh, so cute. I can’t wait to work on both of these new pieces.

An Alternate Ending to Battlestar Galactica

After some five years, Battlestar Galactica ended with a giant two-hour episode. I enjoyed it, though I felt that, in the end, it undermined one of the series’s core themes: it’s individual’s decisions that matter, and those decisions, even if made for what seem like the right reasons, can turn out badly. The road to anywhere is paved with good intentions.

It didn’t have to be that way. I think a few changes to the final episode would have reinforced the theme of people’s individual choices.

Continue reading An Alternate Ending to Battlestar Galactica

Websites that Inspire me to Make things

I’ve mentioned before that I read a ton of craft blogs. Actually that’s misleading, look at the photos on a ton of craft blogs, would be more accurate. This is a list of my favorites that I look forward to seeing every day.

A Print a Day: She does beautiful illustrations and posts every day. On Fridays she has free downloads of her amazing work. I took one of her note page illustrations and made this cute book for Ashley.
Matrioshka Notebook

Creativity Prompt: This is a treasure chest of ideas for making journals as well as creative writing prompts. I have enjoyed the journaling prompts very much and plan my return to paper journaling soon. I have made several of the creative projects. I made three hand-sewn recycled journals for family members for Christmas (that I forgot to take photos of — Ugh!) and this altered book:
Altered Children's Chipboard Book

Elista Mora: Elista does paper cuts and she illustrates, makes miniature dolls, jewelry and clothes. I totally want to grow up someday and have a studio like hers. Mostly, I’d love to dedicate my time to making things as often as it seems she does. Must reprioritize my day!

WhiMSy Love: Nikki is one crafty chick! She always has new things she’s working on and she has two kids that she is initiating into the crafty/creative life. The gal must have a camera embedded in her hand as well. She photographs everything.

Habit: A few friends post snippets of daily life that read like poetry. And photos that are interesting both in the capturing of the moment and for their still life-like quality. I get up in the morning wondering what the gals did that they will post about.

Barrel of Monkeys: Jill is a graphic designer who makes buttons. Sound familiar? She has an huge Etsy shop of very cool buttons. Her design site has some very nice work on it as well. I just found her this week, so I had to share.

Pikaland: The Illustrated Life: I think I must be a closeted illustrator because all the blogs I love are about illustrators. This one has a great twist though. You can buy a PikaPackage that is an assortment of art prints, cards, buttons, ’zines and other goodies from various illustrators. I haven’t bought one yet but you know that’s just around the corner. The art is so very cool that I go to the site even when I know there aren’t any new updates and drool over the stuff I’ve already looked at a half-dozen times.

The Toy Society: This is the most fabulous idea. You make soft toys and then leave them in places to be found by kids. The site has all the info for how to join, how to prepare and package your toy, and ideas of where to leave them once you’re finished. I joined this week and hope to leave a bird to be found very soon. The blog has photos of toys about to be left in the wild and toys that have been found usually with their new owners. Can’t look away!

And lastly, a post that I found this morning looking for the websites I wanted to put in this post. Lately, I’ve been feeling like my craftiness is not that crafty so this was a real pick me up. From Modish Biz Tips: When You Feel Like you Just Don’t Measure Up.

Hoops, Baby, Hoops!

No post from me today, as my brain is too full of basketball to cope. Instead, why not listen to Tor’s mashup of Sufjan Stevens and Blackalicious?

[audio:The Dress Looks Nice On You _ Make You Feel That Way.mp3]

Strawberry Bread

A friend gave me this recipe a couple of years ago. I’ve made it with frozen strawberries a few times but nothing beats fresh berries. You can pretty much make it in one bowl with a wooden spoon which makes it super easy and yummy to boot. The baking time can vary a little bit depending on how juicy the berries are. It’s supposedly from the Portland Palate, 1992.

1 ½ cups all-purpose flour
1 cup granulated sugar
1 ½ tsps ground cinnamon
½ tsp salt
½ tsp baking soda
2 eggs, beaten
½ cup vegetable oil
1 ½ cup fresh strawberries chopped
½ cup chopped walnuts or pecans (I recently used macadamia nuts and it was awesome!)

Preheat oven to 350°.

Grease and flour a 9×5 loaf pan.

In a large bowl, combine flour, sugar, cinnamon, salt and soda. In a separate bowl, combine eggs, oil, strawberries and nuts. Add to flour mixture.

Pour into prepared pan and bake in 350° oven for 50 to 60 minutes.

Lately (In Pictures)

Liza has discovered dress up:
Snow White
Harvest Girl

and dancing on Daddy’s feet:
Stephen, sans head, and Liza dancing one morning before work.

and strawberry bread:
The best thing since...well, you know.

and the fact that big brother is the best thing since the strawberry bread.
A rare moment of togetherness.

Eli has discovered if he helps cook he gets first dibs on licking cake batter:
Slightly posed but still awesome.

and sometimes close is too close to his little sister:
The bucket is only fun for so long.

and that it’s true if you leave your food out, bugs will come and eat it:
Ladybug Picnic

and his favorite storybook characters live at the Children’s Garden at the Huntsville Botanical Garden.