Fix $\alpha\in [0, 1)$ and $m\in\mathbb{N}$. Suppose there exists $n\in\mathbb{N}$ such that there exists a set of complex vectors $\{v_i\}_{1\leq i \leq n}\subseteq \mathbb{C}^m$ that satisfies $\langle\, v_i \,,\, v_i\,\rangle = 1$ for all $1\leq i \leq n$ and $\big\vert\, \langle\, v_i \,,\, v_j\,\rangle \,\big\vert = \alpha$ for all different pairs of $1\leq i, j\leq n$. Can we find the least upper bound for $n$ in terms of $m$ and $\alpha$?
When $\alpha=0$, one obvious example is the standard basis of $\mathbb{C}^m$. In general, the upper bound could be larger than $m+1$ based on this post from MO. Unfortunately, I cannot find ideas from that post helpful. Any hints will be appreciated.
Update: this question is also related to SIC-POVM, a topic that is currently being actively researched
This is a very difficult problem in general. The Welch bounds provide a set of lower bounds on $\alpha$ in terms of the parameters $n,m$ (the standard formulation has $m$ and $n$ switched from yours). The simplex set is one specific solution. More generally what you are looking for is an equiangular tight frame.