Consider unit vectors $|v_i \rangle$ on an $n$ dimensional vector space, which obey the following relation:
$$\langle v_i|v_i \rangle =1 \quad \& \quad |\langle v_i|v_j \rangle| \leq \epsilon, \quad i \neq j.$$
Note that this implies that if $|v_i \rangle = (a_1, a_2 ... a_n)$, then each $|v_i \rangle$ lies on an $(n-1)$ dimensional sphere of unit radius,
$$a_1^2 + a_2^2 + ... + a_n^2 =1$$
If $\epsilon =0,$ the maximum number of such vectors $|v_i \rangle$ with which we can satisfy the inner product conditions is trivially $n$. However consider a finite but a small $\epsilon$.
My Question: I want to demonstrate(/or verify) using Mathematica via some explicit construction that there can be many more vectors than say, $n=700$ which obey this condition if $\epsilon \neq 0$, and if possible, approximately determine the maximum number of such vectors which can be embedded on the $(n-1)$ dimensional sphere.
One way to do this is to start iteratively. The first vector can be at some point
$$|v_1 \rangle = (1, 0 ... 0),$$
Therefore the second vector is will have $a_1^2 \leq \epsilon^2$ using the inner product relation. If we assume equality for the second condition in the inner product. Therefore the second vector will have $a_1^2 = \epsilon^2$ and
$$a_2^2 + a_3^2 ... + a_n^2 = 1-\epsilon^2.$$
Is there a way I can implement this iterative procedure and arrive at the maximum number of vectors for say, $n \sim 700$ and by assuming equality in the second condition of the inner product? I want to demonstrate(/or verify) using Mathematica via some explicit construction that there can be many more vectors than $n=700$ if $\epsilon \neq 0$.
EDIT: Also see my linked question.