Is there a lower bound for the inner product of two complex vectors with unit magnitude entries?

Question

From Cauchy–Schwarz inequality, we know that $\left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \cdot \langle \mathbf {v} ,\mathbf {v} \rangle $, where $\mathbf{u},\mathbf{v} \in \mathbb{C}^N$ and $\langle \cdot \rangle$ denotes the inner product operation. I am wondering whether there are conditions which lead to a lower bound for two complex vectors, when each entry for both vectors has unit magnitude as $\left| \mathbf{u}(i)\right|,\left| \mathbf{v}(i)\right| = 1, i=1\ldots N$.

Answer 1

There are a number of "reverse cauchy-schwarz" inequalities but obviously you'll need some assumptions since the two vectors can be orthogonal in which case the inner product is zero. See this paper, for example.

One particular case is the Diaz-Metcalfe inequality. If $a_k,b_k$ are $n$ complex numbers with $a_k\neq 0,$ for all $k=1,\ldots,n$ and $$ m\leq \mathrm{Re} \frac{b_k}{a_k}+\mathrm{Im} \frac{b_k}{a_k} \leq M, $$ as well as $$ m\leq \mathrm{Re} \frac{b_k}{a_k}-\mathrm{Im} \frac{b_k}{a_k} \leq M, $$ for $k=1,\ldots, n,$ then we have $$ \sum_{k=1}^n |b_k|^2+mM \sum_{k=1}^n |a_k|^2\leq $$ $$ \leq (m+M)\mathrm{Re} \sum_{k=1}^n a_k \overline{b_k} \leq |m+M| \left|\sum_{k=1}^n a_k \overline{b_k}\right| $$