Originally Posted by ZenBeam
This sounds like entangled particles(*), except that as I understand it, once one of the directions is measured, the entanglement breaks. So while the first particle measurement gives X → 1 (for both particles), the measurements for Y and Z are no longer entangled. In short, I guess, I question their assertion that twinned particles, as defined, exist.
* I've always heard that entangled particles have opposite spin, but that distinction probably isn't important here.
It's a little more complicated than this. What you say is correct for the common example of a pair of spin-1/2 particles in the "spin-0 (singlet) state" |↑↓>-|↓↑> (*). A measurement of either particle's spin along the z axis will project the state into either |↑↓> or |↓↑>, so the two particles' spins must be oppositely directed. It turns out that the spin-0 state is rotationally invariant, so the same result actually holds for measurements along any axis.
However, the example given in the paper is for two spin-1 particles. These live in a three-dimensional Hilbert space, not a two-dimensional space like the spin-1/2 particles. This means that measuring the spin of one of these particles along the z axis has three possible results: |↑> (spin +1), |↓> (spin -1), and |0> (spin 0). The measurement proposed in the paper is chosen so that it does not distinguish between the states |↑> and |↓> (this is what it means when they say that it is measuring "the square of the component spin"); so a measurement result of 0 projects onto the state |0>, but a measurement result of 1 projects onto the subspace
spanned by the states |↑> and |↓>. It turns out (as a simple consequence of the quantum-mechanics angular-momentum rules) that for a spin-1 particle, measuring the squares of the x, y, and z components of spins always produces two 1s and one 0. This is their "SPIN axiom" (discussed in their Endnote 1).
Because a single measurement does not fully project the particle state, it does not fully destroy the entanglement. This is what allows the three measurements to all give correlated results. As discussed in Endnote 2, the two-particle state is again the singlet state, a rotationally-invariant state of total spin 0. If the two experimenters actually measured the z-components of spin, they would get opposite results (|↑> and |↓> or |0> and |0>), but since they are only measuring the squared z-component, they always get the same answer.
(*) I'm ignoring normalization.