# Maximum number of "almost orthogonal" vectors one can embed in Hilbert space

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In a Hilbert space of dimension $d$, how do I calculate the largest number $N(\epsilon, d)$ of vectors $\{V_i\}$ which satisfies the following properties. Here $\epsilon$ is small but finite compared to 1.

$$<V_i|V_i> = 1$$

$$|<V_i|V_j>| \leq \epsilon, i \neq j$$

Some examples are as follows.

1. $N(0, d)$ = d

2. $N\left(\frac{1}{2}, 2\right)$ = 3, this can be seen by explicit construction of vectors using the Bloch sphere.

3. $N\left(\frac{1}{\sqrt{2}}, 2\right) = 6$, again using the same logic.

How do I obtain any general formula for $N(\epsilon, d)$. Even an approximate form for $N(\epsilon, d)$ in the large $d$ and small $\epsilon$ limit works fine for me.

EDIT: The question is now resolved. See the answer at https://mathoverflow.net/a/336395/78150

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