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.

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.

The 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.

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.

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.