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openasocket 27 minutes ago [-]
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.
akkartik 4 hours ago [-]
"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.
linuxhansl 4 hours ago [-]
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.
marginalia_nu 2 hours ago [-]
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.
tauwauwau 4 hours ago [-]
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
drdeca 4 hours ago [-]
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.
tauwauwau 3 hours ago [-]
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.
fasterik 26 minutes ago [-]
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
(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.
As a non-mathematician this is a wonderfully evocative summary. I'm curious if mathematicians find it a reasonable characterization of the proof.
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.
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.
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.
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.