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could someone please tell what mistake I am doing by the following gedankenexeperiment.

Imagine two Stern-Gerlach-like devices that measure spin projections of some spin-1/2 particles. Right between them in the middle there is some trap (I don't care about technical details) which contains an entangled state of two identical spin-1/2 particles with a total spin 0. At some point the entangled particles fly apart and go through the detectors, which simultaneously measure the spin projections (yeah, I know, simultaneity is relative, let's say that the measurements happen exactly simultaneously in the center mass system.

But here's the important point. Let's say it all happens on the z-axis (trajectories of the particles, orientation of devices). And let now the first device measure the x-projection of the spin and the second the y-projection!

If there were only one device, measuring, say, the x-projection of the spin of the first particle, then we would know that if we measure, say, "spin down", then we know that the other one is "spin up". Ok.

And now we measure both particles - different projections that do not commute. When we measure the first particle, say, "spin down" in the x-direction then the other one should be "spin up" in x-direction, but we also measure its y-projection of the spin (which gives us the y-projection of the spin of the first particle)!!! This means, that we know BOTH the x- and the y-projection of the spin of both particles. How can it be? The corresponding operators do NOT compute!

What's the point that I am missing? Where is my mistake?

Thanks

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# Gedankenexperiment on quantum entanglement

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