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I think I understand correlations of EPR-state and GHZ-state which deal with spin-1/2 particles and (-1, 1) measured values. Conway's state is spin-1 particle state with (-1, 0, 1) measured values. Which quantum mechanical state vector correspond to Conway's state? And how to prove SPIN and TWIN axioms of Conway's free will theorem from quantum mechanics?

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The TWIN axiom just says that entanglement exists (at any distance), so it is trivial in proper quantum mechanics. The relevant entangled state is of the form $$ |\Psi \rangle = \sum_{m=-j}^j |m\rangle \otimes |-m\rangle $$ or something of this form (phases, sign flips).

The SPIN axiom is also simple in proper quantum mechanics. It says that the eigenvalues of three operators $(J_x^2, J_y^2, J_z^2)$ will be $(0,1,1)$ or some permutation of these three numbers for some states of a physical object (a spin-one particle is enough).

It's easily realized for spin-one particles. In fact, it's enough to represent the spin as the orbital angular momentum and look at the spherical harmonics $Y_{\ell m}$ for $\ell=1$ and $m=-1$ or $0$ or $+1$.

Take $Y_{10}$, for example. It's proportional to $z=\cos\theta$ on the unit sphere. This wave function is clearly annihilated by $J_z$, the generator of the rotations around the $z$-axis. What is the action of $J_x^2$ on it? Well, we know the action of $J^2=J_x^2+J_y^2+J_z^2$ on $Y_{10}$: it is $j(j+1)\hbar^2=2\hbar^2$ times the same wave function.

A few sentences ago, we saw that $J_z$ annihilates $Y_{10}$, so the $J_z^2$ term contributes nothing. By the $x\leftrightarrow y$ reflection symmetry of $Y_{10}$, the two remaining terms $J_x^2$ and $J_y^2$ have to contribute the same i.e. one-half of $2\hbar^2 Y_{10}$ i.e. $\hbar^2 Y_{10}$. That's where the eigenvalues $(1,1)$ for $J_x^2$ and $J_y^2$ come from; the factor $\hbar^2$ was divided by or set to one (choice of units). Even though $Y_{10}$ isn't an eigenstate of $J_x$ or $J_y$, it is actually an eigenstate of their squares. This is analogous to the fact that $\cos kx$ isn't an eigenstate of $i\partial / \partial x$ but it is an eigenstate of $\partial^2 / \partial x^2$.

Similarly, the other two basis vectors of the 3D space of $Y_{1m}$ spherical harmonics, those proportional to $x$ or $y$ on the unit sphere (they are either the sum or the difference of $Y_{1,\pm 1}$, perhaps with phases that may change for unusual conventions), respectively, give the other two permutations of $(1,1,0)$ for the eigenvalues of $J_x^2,J_y^2,J_z^2$.

The theorem and especially its starting axioms are trivial in proper quantum mechanics. They only become non-trivial in various alternative theories people could propose instead of quantum mechanics, namely various hidden-variable theories and GRW-type collapse theories etc. The theorem shows that these hidden-variable and collapse theories cannot be made compatible with relativity.

I think that you meant Conway's and Kochen's axioms.

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