Minimum Value Of a Complex quantity Let $Z_1,Z_2,Z_3,Z_4 \in\mathbb{C}$ such that 
$Z_1+Z_2+Z_3+Z_4=0$;$|Z_1|^2+|Z_2|^2+|Z_3|^2+|Z_4|^2=1$ then the minimum value of $|Z_1-Z_2|^2+|Z_2-Z_3|^2+|Z_3-Z_4|^2+|Z_4-Z_1|^2$ is ?
 A: 
Just note that $(z_{1},z_{2},z_{3},z_{4})\in\mathbb{S}^{3}$ (the unitary sphere in $\mathbb{C}^{4}$) so any permutation of its entries lies in 
  $\mathbb{S}^{3}$ then immediately we can see 
  $$|z_{1}-z_{2}|^{2}+|z_{2}-z_{3}|^{2}+|z_{3}-z_{4}|^{2}+|z_{4}-z_{1}|^{2}\le 2 ~~~(*)$$
  But easily we can see that  in $(*)$ the equality can hold. Indeed thake 
  $(1,0,0,0).$ An way more technical to solve this problem is note that 
  the 'operation' $(x,y,z,w)\mapsto (y,z,w,x)$ is a rotation of an angle 
  $\theta=\pi/3$. 

A: $$|Z_1-Z_2|^2+|Z_2-Z_3|^2+|Z_3-Z_4|^2+|Z_4-Z_1|^2 = 2\times(|Z_1|^2+|Z_2|^2+|Z_3|^2+|Z_4|^2)-(Z_1\bar{Z_2} + \bar{Z_1}Z_2 + Z_2\bar{Z_3} + \bar{Z_2}Z_3 + Z_3\bar{Z_4} + \bar{Z_3}Z_4 + Z_4\bar{Z_1} + \bar{Z_4}Z_1)$$ 
$$ = 3\times (|Z_1|^2+|Z_2|^2+|Z_3|^2+|Z_4|^2)- \left(|Z_1|^2+|Z_2|^2+|Z_3|^2+|Z_4|^2 + 2 \Re(Z_1\bar{Z_2}) + 2\Re(Z_2\bar{Z_3}) + 2\Re(Z_3\bar{Z_4}) + 2\Re(Z_4\bar{Z_1}) + 2\Re(Z_1\bar{Z_3}) + 2\Re(Z_4\bar{Z_2}) \right) + 2\left( \Re(Z_1\bar{Z_3})+\Re(Z_4\bar{Z_2}) \right) $$
$$=3\times (|Z_1|^2+|Z_2|^2+|Z_3|^2+|Z_4|^2)- |Z_1+Z_2+Z_3+Z_4|^2+2\left\{\Re(Z_1\bar{Z_3})+\Re(Z_4\bar{Z_2})\right\}$$ 
$$= 3+2\left\{\Re(Z_1\bar{Z_3})+\Re(Z_4\bar{Z_2})\right\}$$
Next Step:
$$=2+\left\{|Z_1|^2+|Z_2|^2+|Z_3|^2+|Z_4|^2+2 \times \left(\Re(Z_1\bar{Z_3})+\Re(Z_4\bar{Z_2})\right)\right\}$$
$$=2+\left(|Z_1+Z_3|^2+|Z_2+Z_4|^2\right) \geq 2.$$
Equality holds when, 
$$Z_1 = -Z_3 \text{ and }Z_2 = -Z_4 \text{, also } |Z_1|^2+|Z_2|^2=\frac{1}{2}.$$
So, I think this is the most tightest lower bound (not fully sure about it) we can give for the given expression based on the given constraints.
A: Notation: $Z\cdot W=\frac{z\bar{w}+\bar{z}w}{2}$ 
therefore 
$|\sum Z_i|^2=\sum Z_i \sum \bar{Z_i}=\sum|Z_i|^2+2\sum_{i\neq j} Z_i\cdot Z_j \implies \sum_{i\neq j} Z_i\cdot Z_j=-\frac{1}{2}$
$|Z_1-Z_2|^2+|Z_2-Z_3|^2+|Z_3-Z_4|^2+|Z_4-Z_1|^2=2(1-\sum Z_i\cdot Z_j)+2Z_1\cdot Z_3+2Z_2\cdot Z_4=3+2Z_1\cdot Z_3+2Z_2\cdot Z_4=2+|Z_1+Z_3|^2+|Z_2+Z_4|^2\ge 2$
Equality $\iff z_1+z_3=z_2+z_4=0$
