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.


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