Consider $N$ spin-1/2, for which we can define the collective spin operator $\vec{S}=\sum_i \frac{\vec{\sigma}^{(i)}}{2}$. My question is, what is the upper bound $U$ on $$ f(\rho) = \text{Var}[ S_z ] - \langle S_x\rangle $$ for separable states $\rho$? Meaning $f(\rho)\leq U$ for all $\rho$ separable, and with the bound saturable by some $\rho$. (In the above, $\text{Var}[ S_z ]=\langle S_z^2\rangle-\langle S_z\rangle^2$ is the variance).
A naive conclusion would be to bound each quantity independently, e.g. based on the spectrum of the operators, $\Delta^2 S_z\leq \frac{N^2}{4}$ and $S_x\leq \frac{N}{2}$, but I think $U^\prime=\frac{N^2}{4}+\frac{N}{2}$ cannot be attained by separable states. (One has also to fulfil the uncertainty relation).
Therefore, is there a convenient approach to tackle this kind of question?
