# Using the upper bound lemma, show that $\|Q-U\|\geq\frac{1}{2|G|}\sum^*d_{\rho}Tr(\hat{Q}(\rho)\hat{Q}(\rho)^*)$

Using the upper bound lemma: Let $$Q$$ be a probability on the finite group $$G$$. Then, $$\|Q-U\|^2\leq\frac{1}{4}\sum^*d_{\rho}Tr(\hat{Q}(\rho)\hat{Q}(\rho)^*)$$ where the sum is over all non-trivial irreducible representations.

Show that $$\|Q-U\|\geq\frac{1}{2|G|}\sum^*d_{\rho}Tr(\hat{Q}(\rho)\hat{Q}(\rho)^*)$$ and $$\|Q-U\|^2\geq\frac{1}{4|G|}\sum^*d_{\rho}Tr(\hat{Q}(\rho)\hat{Q}(\rho)^*)$$

I tried to use the simply formula that defines $$\|Q-U\|$$ to show the bounds, but I'm keep running into troubles. Any help is appreciated.