Hardy’s Paradox

I just read about a crazy quantum mechanical thought experiment from 1992 called Hardy’s Paradox, wherein electrons and positrons can essentially undergo incomplete annihilation. As far as I can tell, it’s not quite a true paradox: it’s a prediction of quantum mechanics that is perfectly mathematically consistent, it just happens to completely defy our physical intuition. I’ll provide my attempt at a summary of the paradox below, and describe a ridiculous experiment that has just been published which offers a direct observation of the effect. If you want a more thorough explanation of Hardy’s Paradox, I refer you to “The Mystery of the Quantum Cakes”, an article (available free here, for some reason) aimed at an audience of science teachers.

The experimental setup can be seen clearly in the terrible image (left), which took me a good thirty minutes to decode, or in the better image (right) produced by some nameless intern with both physics and graphic design skills. As the caption in the latter image suggests, the experimental setup here involves electrons, positrons, beam splitters, and detectors – the standard quantum mechanical toolset.f2 If you just want a cursory explanation of the experiment, here’s the article from where that image came. If you want a deep understanding of the experiment, the relevant papers are here and here. I’m going to aim for a middle ground and try to explain it at a level appropriate for someone with a qualitative feel for quantum mechanics (i.e. familiarity with terms like “wavefunction” and concepts like the two-slit experiment), but no formal training or classes. This will basically involve me explaining the diagram and alluding to some equations in the papers – if you already understand the equations, then you can just read the papers yourself. Incidentally, in everything to follow, the + and – signs correspond to paths and detectors for the positively-charged positrons and for the negatively-charged electrons, respectively.

Take a stream of electrons and a stream of positrons, and send each through its own beam splitter, so each particle’s wavefunction splits into components along path V and along path W. Both paths V and W end up at a second beam splitter, which is tuned so that the two components of each particle’s wavefunction meet up in phase and are guided towards detector C. So, if both streams are completely separate and if we don’t observe the particles as they move through the experiment, all the particles will be detected at their respective detectors C. Now, bring the two streams together so that path W+ (the one with positron wavefunctions) crosses path W- (the one with electron wavefunctions).

In this case, the positron wavefunction in W+ can interfere with the electron wavefunction in W-, leading to a variety of new behavior. For example, this interference can change the phase of the electron wavefunction at the second beam splitter so that there is a chance of it being diverted to the D- detector instead of proceeding as usual to the C- detector. In the same way, the presence of the electron’s wavefunction can send the positron to D+ instead of C+. Thus, we can think of the D detectors as signals that the particles have interacted, i.e. a detection in D- means that the positron went through path W+ and affected the electron’s wavefunction, and a detection in D+ means that the electron took path W- and affected the positron’s wavefunction. Of course, if both particles take paths W, then they will meet and annihilate each other. Thus, one would expect it to be impossible to see both the positron and the electron end up at detectors D. Unfortunately for my intuition, it’s not impossible – there’s a 1/16 chance of it! The derivation is equation (1) in the first of the two papers, and while I can follow the math, I haven’t been able to come up with a good physical explanation for why this is possible. My best guess based on the results in the subsequent pargraphs, is something to do with entanglement and adding fractions of the states with one particle interfering with the other.

Anyway, Irvine et al. ran a version of this experiment using photons instead of electrons and positrons, and they found the predicted amount of D-D+ events, which was greater than zero by twelve standard deviations! Even more intriguing and confusing is this plot from the paper by Yokota et al. (2009). They also used photons in a similar experimental setup, but they also placed detectors in the middle of the paths to observe the particles (photons) as they moved through the experiment. These observations should collapse the wavefunctions and invalidate the experiment, so they opted to use “weak observations”, where the detector only observes each particle to a fraction of the precision necessary to measure its state. Adding together a few thousand identical weak observations should produce enough signal to piece together the underlying measurement without disturbing any single particle enough to collapse its wavefunction.

This sounds really dubious to me in theory, but Figure 4 of their paper (reproduced below) is pretty compelling. The horizontal axis is the strength of the measurement (zero is no observation, one is a full observation, and in between is a “weak observation”). The vertical axis is the probability of finding a particle in a given state, assuming the final outcome is a spooky D-D+ detection. Note that the uncertainties increase as the observation gets weaker, and the probabilities at each measurement strength add up to one. From top to bottom on the left of the plot, the line with triangles is analogous to the electron taking V- and the positron taking W+, the line with circles is analogous to the electron taking W- and the positron taking V+, the line with squares is analogous to the electron taking V- and the positron taking V+ (the non-interference case), and the line with diamonds is analogous to the electron taking W- and the positron taking W+ (the annihiliation case).

So, there are basically no annihilations if you have a D-D+ detection, which makes sense. Instead, these detections seem to be composed of equal parts of the two interference cases. And, as the observation strength diminishes, the probability of each of these cases goes to one – so both are happening at once! Additionally, the probability of the non-interference case goes to -1, which of course makes no sense to me – I guess it’s just very improbable!

I wish I could provide a grand summary of Hardy’s paradox, but all I can offer is a smattering of comments instead. As both papers note, the experiments are further refutations of local hidden variable theories of quantum mechanics that posit an underlying “normal” reality to which the probability theory is only an approximation. That’s nice, but these theories have already been discredited by other experiments. I am more interested in understanding what it means to have a D-D+ detection, and I am most interested in understanding what is implied by that scaling of probability with measurement strength. I know some of my readers are better at physics than I am, so I’d welcome any comments or ideas if you’ve got em!


One Response

  1. Is this like a little Zeno moment spread wide?

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