You've forgotten that the point is not to make the position fuzzy. The point is for the far away detector to register a particle in a definite position. (In one interpretation, the detection process collapses the wave function, albeit how it does this has never been explicated.)When the far away detector registers a particle, it decodes it as "yes" or "1." Until it detects the particle, there is no signal.
The probability of the detection of a particle at any point is equal to sum of the squares of the amplitude of the entire wave function. The probability of detection of the particle at site of the far away detector at the time it was "sent" (ignore relativity, please,) is extremely low. This makes sending signals that way impossible. Over time the wave function evolves, until there is a high probability of detection by the far observer. This occasion will be the at the time it would have taken the particle to travel through space normally.
A quantum particle's momentum/position cannot be measured with a certainty that goes beyond Heinsenberg's uncertainty minimum (h/2pi). That's the lower limit of the fuzzyness of a quantum particle - under NO circumstances will the particle ever be less fuzzy than this.
Above this limit, a quantum particle is free to become fuzzier, and, indeed, it becomes fuzzier maturally, as the time passes. That's the wave function evolution you talked about - the probability of detecting the particle in distant positions increasing over time.
Once a particle's fuzzyness respects Heisenberg's uncertainty, yes, its wave function evolves and the probability of it being detected at the receiver increases ONLY over time.
But if a particle's fuzzyness does NOT respect Heisenberg's uncertainty (because its momentum is too precisely measured) its position becomes highly fuzzy instantaneously and not 'over time'.
Strictly speaking, any talk of trajectories is incorrect. All wave functions are infinite. The wave packets that appear to us as particles disperse, so there are no paths taken by any elementary particle. (If this seems odd, think about orbitals in atoms.) In principle any particle could travel superluminally. It is not clear to me how measuring the momentum of a particle (even if such ultraprecise measurements are even physically possible) would increase the reach of an already infinite wave function.
Wave functions evolve over time respecting the speed of light barrier (if we exclude improbable solutions).
Wave function collapse, on the other hand, is more interesting from this POV - as Einstein called it, 'spooky action at a distance' aka entanglement - and this is instantaneous (this 'instantaneity' is not improbable; it happens every single time).
even if such ultraprecise measurements are even physically possible
In principle, there's nothing that prevents measurements to be so untraprecise.
In practice, though, we're FAR from such a precision.
In terms of the double slit experiment, the detector on the far side of the two slits always detects a single electron. It is only after a period of time that the interference pattern is revealed.
This is correct.
But this electron is detected at one of the interference fringes aka it interferes with itself.
The electron is NOT detected at the 2 finges corresponding to the two slits.
Incidentally, I believe that in the usual terminology quantum interference is not a superposition. I think "superposition" references the need for a combined wave function to describe the evolution of a quantum system.
Generally speaking, 'superposition' refers to a quantum particle simultaneously having two properties that, in classical physics, are cotradictory.
Quantum interference via the two slit experiment is due to a quantum particle being in two places at once - hence the use of 'superposition'.
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