Graduate student proves a quantum uncertainty principle for fractals | Hacker News Reader

Graduate student proves a quantum uncertainty principle for fractals

quantamagazine.org

71 pointsbookofjoe12 comments

"In 2017, Dyatlov and Long Jin from Tsinghua University in Beijing used the one-dimensional fractal uncertainty principle to prove that you can never trap a wave on a hyperbolic surface; it will always spread out until it touches every corner. To do so, they imagined a region on the surface that a wave never enters, even after having infinite time to spread out. When they removed all the trajectories that entered that region, what remained was the same sort of fractal dust that appeared in the pinball example. Since the fractal uncertainty principle forbids a wave from being trapped on a fractal, no such region can exist — the wave must spread everywhere."

As a non-mathematician this is a wonderfully evocative summary. I'm curious if mathematicians find it a reasonable characterization of the proof.

As a physics layman I find it fascinating how quantum mechanics are tied to information theory.

For example, take quantum decoherence (which, IMHO, is the most logical explanation for the collapse of the wave-function - by saying it does not actually collapse). Quantum decoherence is almost like a giant constraint resolution system - once a particle randomly interacts with another they become entangled and both now have fewer degrees of freedom. When it interacts with many particles, like any macro-effect it has essentially no degrees of freedom anymore. It's all about who knew about what and when. The experiments around this fascinating. (Note that there are other theories, like the many-worlds interpretation, that also explain the collapse of the wave function)

This seems to be another example of this. Anyway, as I said, just a layman.

Doesn't entanglement mean that entangled particles just cannot have same state of the entangled quantum property at the same time, but they can still achieve all states, essentially preserving their degrees of freedom

No.

A state is entangled when it isn’t a product state.

Two spin (1/2) particles in a singlet state have the kind of “they have opposite states” thing going on that you describe, and is a specific way that two particles can be entangled.

OK, so instead of having all states (00, 01, 10, 11) available in entangled state they only have 01 and 10 available because they have to be opposite of each other, but even with that these particles individually are able to have both states right? I'm not knowledgeable in this field, I just have interest.

00, 01, 10, 11 are separable states, meaning that a pair of particles in one of those states can be described as two separate one-particle systems. For example, 01 means that the first particle is in the state 0 and the second particle is is in the state 1.

A state like (01 + 10) is not separable, so by definition it's an entangled state. "Separability" is a straightforward algebraic fact that follows from the definition of a vector and the tensor product. You can see what this means in the following Google answer

https://share.google/aimode/13jNpR7bmpPMo1pn3

(01 + 10) means that if I measure the first particle and get 0, then the second particle is now in the state 1. If I measure the first particle and get 1, then the second particle is now in the state 0.

The state you describe, sqrt(1/2) ( |01> + |10>) is an entangled state, but not all entangled states are like that.

The state sqrt(1/2) ( |00> + |11> ) is also possible, and is also an entangled state, but doesn’t have the two particles in opposite states.

By contrast, the state (1/2) (|00> - |01> + |10> - |11>) is (while a valid state) not an entangled state, because it is equal to (1/2) (|0> + |1>) (|0> - |1>) .

If quantum probability is a wave function, and interaction introduces a phase shift, this alone is enough to lead to decoherence through the same mechanics as classical optical (de)coherence.

In the same circumstances a light beam stops producing an interference pattern in the Young experiment, quantum wave functions do as well. This is pretty easy to derive, just introduce a random phase shift term, and average across it, and the interference pattern disappears and a bell curve emerges instead.

For those that aren't aware, the uncertainty principle goes pretty deep. You can actually define the uncertainty principle as an inequality involving the integral of a function vs the function's Fourier transform: https://en.wikipedia.org/wiki/Uncertainty_principle#Harmonic... . And in a quantum system you can construct the momentum of a particle as the Fourier transform of the position (up to a constant) and the uncertainty principle falls out because of this. This article doesn't state that super explicitly. So the real interesting thing being done here is getting a Fourier transform that works on fractal spaces.

Another phenomena that boils down to this is the diffraction limit in optics. I.e. the far-field diffraction pattern produced by a point source (c.f. point spread function) is the spatial Fourier transform of the aperture shape. Thus the aperture and the point-spread function satisfy a similar relation to the momentum-position uncertainty principle.

I'm not very good at math, and I read quantum physics in lay-books, but I would like to say that whoever wrote the article did a good job leading to where it was going (even if maybe not clear to me), I ended up thinking "Square root of -1" and then a few paragraphs later, there it was...

This could have just been a lucky guess, but I feel like the way they presented it my intuition understood what was happening though I can't mentally see/grok it. I think making music and messing with waveforms in synths also helps to understand this intuitively more than the actual math.

I would like to credit "The Theory of Almost Everything" by Robert Oerter for explaining it in a way I could understand 15 years ago https://www.goodreads.com/book/show/183207.The_Theory_of_Alm...

Great book with the history and explanation of a lot of quantum stuff, without being huge like Brian Greene's books (which I also love).

>Isolating each peak required constructing a very specific function that, when multiplied by the original fractal-like function, would pull out just the peak and be close to zero everywhere else. This is called a

damping function

, and it needs to be perfectly tailor-made to work. “This is a challenging thing to construct,” Cohen said. But he knew that if he could do it in higher dimensions, he could unlock the entire proof."

Brilliant, absolutely brilliant!

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