# Entanglement and the uncertainty principle

Suppose you have two maximally entangled particles. You measure the spin about the x axis of the first and the spin about the y axis of the second. But both spins cannot be known simultaneously, so how is this problem resolved?

## 1 Answer

When someone says that spin measured about different axis can't both be known, they mean that whatever state you pick will have variability in at least one of the possible spin measurements you can do.

So that is what you will get when measure the spin, you will get variable results. This happens even with entanglement with even just one particle. With just one particle, if you first "measure" the spin about the x axis and then "measure" the spin about the y axis. Then you will get variable results for the y axis result. Since this case is more basic, let's make sure we understand it first before we get to the case with entangled particles.

By variable results I mean you get different results different times if you set up the experiment the same way repeatedly. How do you do that? First you need to identically prepare a bunch of particles the exact same identical way (to have lots of particles to have some way to get results that can change or be the same).

Then you must "measure" the spin about the x axis for a particle. Then after you've "measured" the spin about the x axis for that particle and gotten a certain result follow up that "measurement" with a follow up "measurement" on the same particle, this time measuring the spin about the y axis. No entanglement, just measuring about the x axis then measuring about the y axis.

The measurement about the y axis will give us variable results. By that I mean that if we take that huge colelction of identically prepared systems, some of them give us one result for the measurement about the y axis and some give us other results for the measurement about the y axis. In fact every possible result will occur equally frequently. That's what we mean by variable results. We set up the experiment identically as best we could but got different results different times.

This would not have happened if we measured about the x axis on the same particle two times in a row, then we'd get the same result both times, so the results would not be variable. That's what it means about "knowing the spin about two different axis simultaneously". The "simultaneous" is that we can prepare it so that the x axis results have low variability or we can prepare it so that the y axis results are low variability, but no matter how we set up the experiment at least one will have some variability.

I must emphasize this again. The idea of simultaneity was back at the moment when we choose how to set up the experiment. We could have set it up in a way that gave low variability results for measurements about the x axis. Or instead we could have set up the experiment so that we got low variability results for measurements about the y axis. But back when we planned the set up we can't do both.

So know we know what it means that "both spins cannot be known simultaneously." It it means that at least one spin will produce variable results. And we know what it means to have variable results, it means that it we set up multiple instances of the entire experimental set up then we get different results on different set ups.

So now we are ready to bring up entanglement. Things are trickier for entangled particles because the results are already maximally variable for any possible measurement. Yup. When particles are entangled, the property that is entangled produces every possible result available to it and in such a way as to have every possible result appear equally frequently, that's as variable as it can be. Entanglement means that each particle gives highly variable results.

So, each of those results (both measurements about the x axis and measurements about the y axis) give you every possible result for a spin measurement, and each result happens equally frequently (in the heads-and-tails sense of not favoring any one side in the long run, not that it strictly alternates). The entanglement means the results are correlated. For instance if you measure both spins about the x axis you get each possible result equally likely but you are 100% likely to get the results to be the same (if they are entangled that way).

Each side looks as boring as can be, you can pick any axis (not just x,y, or z, but any direction in space) and you get every possible result, each result equally often. The only thing weird is that later when you and your friend compare your results you'll notice that any time you both picked the same axis you got the same result.

Spooky.

• Thank you, this is much more clear. So if I understand correctly, if any particle is maximally entangled with another particle, then when you measure the spin of that particle (and only that particle) about any axis, it must always have a 50-50 chance of being up or down? – Sheepman May 4 '15 at 3:27
• @SamuelMontgomery For spin one half if it is the spins that are maximally entangled, you are correct, 50-50 (for spin 1, its is 1/3-1/3-1/3 chances for being +1,0, or -1, for spin 2 it is 20-20-20-20-20 for being +2, +1, 0, -1, or -2). All for any axis. – Timaeus May 4 '15 at 3:32