# Bounding the value of a function for separable spin states

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?

• There was a typo in the sign, yes. I meant to put just upper bounds based on the operator's spectrum.
– m137
Mar 11, 2021 at 20:59
• Is there a motivation behind this question? (I'm asking since often there might be a better answer for the original problem.) Mar 11, 2021 at 21:20
• This doesn’t make sense unitwise as you are comparing something with units of $\hbar^2$ with something with units of $\hbar$/ Mar 12, 2021 at 0:58
• @ZeroTheHero Good point, which links back to the question about the motivation behind wanting to show such a bound (there might be scenarios if these are not actual spins - though one would indeed expect that for any physical scenario, sigma would be a unit-ful operator.) Mar 12, 2021 at 14:59
• I would check out spin-squeezing criteria for separability/entanglement, there must be something in this direction. Jun 23, 2021 at 0:32