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Currently I am working with finding the solutions for the following problem:
I have a unit sphere in which I have n points defined by their polar and azimuthal angles: $\theta_n , \phi_n$. I then do the inverse stereographic projection to find the corresponding points in the complex plane: $z = \frac{x_1+ix_2}{1-x_3}$ where $x_1=cos(\theta)sin(\phi)$, $x_2=sin(\theta)sin(\phi)$, $x_3=cos(\phi)$, and then use this complex numbers as roots for a polynomial of degree n.

After this I use said polynomial to construct a quantum state $\lvert \psi\rangle=c_1\lvert+j\rangle$+...+$c_{2j}\lvert-j\rangle$, where the coefficients multipliying the base states are the coefficients of the polynomial constructed before.

I am then trying to get the roots for the following function: $\langle S_x^2\rangle+\langle S_y^2\rangle+\langle S_z^2\rangle$, this is going to be a function of 2*n variables and I have no idea how to approach this with numerical methods.
I would be really thankful if someone with more experience in the subject could help me with just and idea of whatever I would have to research in order to approach this.

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  • $\begingroup$ I wonder if it helps to redo the function in terms of $S_z$ and the raising and lowering operators $S_x\pm iS_y$. $\endgroup$ – Carl Brannen Apr 8 at 7:19
  • $\begingroup$ can you please clarify what you mean by $\langle S_x^2\rangle$. Is $S_x^2$ sanwitched between two states $|\psi \rangle$ given in the question?. $\endgroup$ – David Bar Moshe Apr 10 at 9:32

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