Suppose we have an hermitian matrix $H$ with eigenvalues $h_i$, and I want to get the minimum value of its variance, i.e. $\vec{v}^{\dagger}H^2\vec{v}-\left( \vec{v}^{\dagger}H\vec{v} \right) ^2$. How can we rigorously show this?
I tried Lagrange's multiplier method which I think nearly succeed. By spectral decomposition, we can write $H$ as $H=\sum_i\lambda_i|i\rangle\langle i|$(let me use $|i\rangle$ to stand for column vectors while $\langle i|\equiv|i\rangle^\dagger$). Then the problem changes into minimize $$\sum_i{{h_i}^2}\langle v|i\rangle \langle i|v\rangle -\left( \sum_i{h_i}\langle v|i\rangle \langle i|v\rangle \right) ^2.$$ over unit vectors $|v\rangle$.
By replace $\langle v|i\rangle \langle i|v\rangle $ with $p_i$, we finally state the problem as $$ \underset{p_i}{minimize}\sum_i{{h_i}^2}p_i-\left( \sum_i{h_i}p_i \right) ^2 \\ s.t. \sum_i{p_i=1} $$ Now we can start Lagrange's multiplier method. By introduce $\lambda$, we have the Lagrangian is $\sum_i{{h_i}^2}p_i-\left( \sum_i{h_i}p_i \right) ^2+\lambda \left( \sum_i{p_i}-1 \right) $, by derivative w.r.t. $p_i$ we have the linear equation: $$ {h_k}^2-2\left( \sum_i{h_ip_i} \right) h_k=\lambda ,\forall k \\ \sum_i{p_i}=1 $$ We then solve those equations and maybe it might have some easier results by noticing that $\lambda$ only has one value which does not change with different $h_k$.
For example, if we have $p_1$ to $p_3$, we can solve this by writing it in matrix form: $$ \left( \begin{matrix} 1& 1& 1& 0\\ 2h_1h_1& 2h_2h_1& 2h_3h_1& 1\\ 2h_1h_2& 2h_2h_2& 2h_2h_3& 1\\ 2h_1h_3& 2h_2h_3& 2h_3h_3& 1\\ \end{matrix} \right) \left( \begin{array}{c} p_1\\ p_2\\ p_3\\ \lambda\\ \end{array} \right) =\left( \begin{array}{c} 1\\ {h_1}^2\\ {h_2}^2\\ {h_3}^2\\ \end{array} \right) $$ Is there some easier way to solve it? I can't directly see the answer and using Gaussian elimination seems too complicated for more variables.
We don't need to solve the linear equations totally but do mind one important truth: $\lambda$ only has one value while we have several restrictions. Keep this in mind and mind that ${h_k}^2-2\left( \sum_i{h_ip_i} \right) h_k=\lambda $ is parabola w.r.t. $h_i$, hence we can only allow two $p_i\neq0$, or only one. Then it's easy to show that the optimal case is when $p_{max}=p_{min}=\frac{1}{2}$.