Let us build entanglement from the ground up, starting with wavefunctions. Here is an example of a wavefunction:
Another thing you can tell by looking at the wavefunction is the particle's momentum (mass times velocity). The momentum of the particle is related to how wavy the wavefunction is. For instance, if the wavefunction looked like this...
But particles don't just move to the right and left. They can go right, left, forward, backwards, up, and down. We live in a three-dimensional world. The above wavefunctions only show one of those dimensions. Here is an example of a two-dimensional wavefunction:
But "left" and "right" are relative. If we just turn around 180 degrees, left becomes right, and right becomes left. If we turn around 90 degrees, the X and Y coordinates switch places. And if we turn 45 degrees...
Let's say we tried to measure the Y position of the particle, and got something to the left of center, right where that big hill is. Then we already know that the X position will probably be to the left of center too, before we directly measure it. And if we found that the Y position was to the right of center, we could predict that the X position would also be to the right of center. The X and Y coordinates of the particle are correlated. I might even say that the X and Y coordinates have a "mysterious connection" between them.
What makes this mysterious correlation so special? Why do we need to say that the particle is in both places at once? Couldn't we just say that we don't know its precise location? Couldn't we just say that there's a chance that both X and Y are right of center, and a chance that they're both left of center? The answer is no. The particle is in both locations at once, until we measure it. Recall that the momentum of the particle is described by the waviness of the wavefunction. If the particle were only in one of the two locations, that would affect its waviness, so to speak, thus affecting its momentum.
That's the basic idea of how we add new dimensions to quantum mechanics. We take what's called the Cartesian product of the X axis and the Y axis, and we get the X-Y plane. We can take another Cartesian product with the Z axis to make it three dimensional.
When we look at a quantum mechanical system with two particles, it is not described with two separate wavefunctions. There is always only one wavefunction. Instead, we take the Cartesian Product of the two particles' positions, like so.
Of course, each particle really requires not one, but three dimensions to describe its position. So for two particles, we really need six dimensions to describe them both. If we have three particles, we need a 9-dimensional space. If we have thousand particles, we need a 3000-dimensional space. In principle, the entire universe would be described with a single wavefunction with an absurdly huge number of dimensions.
And that's not even including non-spatial dimensions, which also have to be multiplied in. Non-spatial dimensions include spin and polarization. In fact, most popular explanations of quantum entanglement start out with correlations of polarization between two photons. It's the same idea, really. The main difference is that there are only two polarizations (horizontal and vertical), but there are an infinite number of positions (ie 1, 1/100, pi, 42). That makes it much easier to analyze and study under controlled conditions.
Let's take a step back and see what happened here. If we have a multi-dimensional wavefunction, the different dimensions can be correlated with each other. Adding new particles is just a matter of adding new dimensions to the wavefunction. Therefore, separate particles, too, can be correlated with each other. These correlations can extend across large distances. Measuring a particle here can instantly and indirectly measure a particle on the other side of the galaxy. It's almost, but not quite, as if you received a message from the other side of the galaxy at a speed far faster than light. The consequences of entanglement, even I'll admit, are quite profound and mysterious. But where entanglement came from is not mysterious. It was just quietly encoded in the math all along.
1 comment:
Hey, thank you for this post, really. It's always good to have things a bit more clear in your mind :)
I'm catching up with your blog posts, by the way. I had to put it aside due to lack of time in this last month :P
I'll propagate this one in the internet, by the way. I found it very useful.
Regards!
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