Speed of entangled particle changes

Question

If two particles are entangled at a distance of light years, does changing one affect the other instantaneously, or does it take years to occur? What gives us the answer to this question?

Answer 1

If two particles are entangled at a distance of light years, does changing one affect the other instantaneously, or does it take years to occur?

It isn't like each particle had its own individual state and that you could change one and by changing it change the other.

What you had is a joint state of the combined system. For instance if you have entangled spins then the individual particles don't have their own spin direction. And you can put a particle inside a Stern-Gerlach device and the particle will spatially split and the spin for each spatially split part becomes polarized in the direction associated with the Stern-Gerlach device. So it didn't have a polarization before you interacted and now it does. But this affects both particles and you couldn't control how it split.

Let's be concrete. If you have two particles, both spin 1/2 and they are in the state $$\frac{\sqrt 2}{2}\left|\uparrow\downarrow\right\rangle-\frac{\sqrt 2}{2}\left|\downarrow\uparrow\right\rangle.$$

Then to measure the spin you send in a beam and split it. So for instance if your beam moves in the $\hat y$ direction and has some width in the $x$ direction, say from $x=-1$ to $x=+1,$ then the beam widens in the $x$ direction as it moves in the $\hat y$ direction and eventually splits into two beams. Like the letter Y. And the spin changes so that left branch becomes $\left|\uparrow\right\rangle$ and the right branch becomes $\left|\downarrow\right\rangle.$

So you can imagine it as a movie: first, a line segment of length 2 travels, then it gets wider, then splits in two, then each part moves apart, each getting a well defined spin for each branch. It's the letter Y just as a time series of horizontal slices.

Now we can talk about what happens when two particles have their spins entangled. Since there are two particles the wave lives in a 6d space $(x_1,y_1,z_1,x_2,y_2,z_2)$ with $(x_1,y_1,z_1)$ saying a configuration of the first particle and $(x_2,y_2,z_2)$ saying a configuration of the second particle. So saying the configuration of both particles tells you the point in 6d and telling the point in 6d tells you the configuration of both particles. So if they are being split in the x direction we can write both beams by a square with one direction being $x_1$ and the other being $x_2.$

If you put the first particle's beam into a Stern-Gerlach device the square widens, splits along a vertical seam, and separates. As the joint spin state changes from $\frac{\sqrt 2}{2}\left|\uparrow\downarrow\right\rangle-\frac{\sqrt 2}{2}\left|\downarrow\uparrow\right\rangle$ everywhere in the square into being $\frac{\sqrt 2}{2}\left|\uparrow\downarrow\right\rangle$ in the left square and $-\frac{\sqrt 2}{2}\left|\downarrow\uparrow\right\rangle$ in the right square.

Later if you put the second particle's beam into the Stern-Gerlach device you deflect the left square up because it is already that spin and you deflect the right square down because it is already that spin.

If you instead first put the second particle's beam into the Stern-Gerlach device the square lengthens, splits along a horizontal seam, and separates. As the joint spin state changes from $\frac{\sqrt 2}{2}\left|\uparrow\downarrow\right\rangle-\frac{\sqrt 2}{2}\left|\downarrow\uparrow\right\rangle$ everywhere in the square into being $\frac{\sqrt 2}{2}\left|\uparrow\downarrow\right\rangle$ in the upper square and $-\frac{\sqrt 2}{2}\left|\downarrow\uparrow\right\rangle$ in the lower square.

Later if you put the first particle's beam into the Stern-Gerlach device you deflect the upper square left because it is already that spin and you deflect the lower square right because it is already that spin.

In either order you have a joint state where the beams were deflected in opposite directions and now have individual spins states and the individual spin states are opposite.

You could even put both into Stern-Gerlach devices at the same time and they just move to become two squares, one in each corner (upper left corner and lower right corner) and the spins become aligned with the direction the particles went.

Neither experiment can tell which happened first. And you didn't make any particular result happen, you just split the beams into different beams, each with a well defined spin for the individual particles.

So if you had two beams for the two particles the beams get split and then deflected or both and in the end you have spins that are correlated for each region.

What gives us the answer to this question?

The Schrödinger equation when applied to the combination of the object and the experimental apparatus is what predicts this. And the Schrödinger equation seems to be correct for every object and every kind of equipment apparatus we have used it for.