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There is no general exact formula for $N(\epsilon, d)$ and some special cases (for example SIC-POVM) is an area of active research. However there is a Welch bound that gives $\epsilon^2 \ge \frac{n-d}{d(n-1)}, n=N(\epsilon, d)$ and hence bounds $N(\epsilon, d)$ from above.


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This seems like it should be a known mathematical property of Hilbert spaces, but I can't immediately lay my hand on any such result. In lieu of that, this is very far from an answer to your question, but it perhaps indicates the difficulty of (some of) what you're asking... First, perhaps we can clarify your problem statement. I assume you mean $$ |\langle ...


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