Suppose we have n normalized vectors on an arbitrarily large Hilbert space $|A\_1\rangle,\dots,|A\_n\rangle$, $\langle A\_i|A\_i\rangle=1$ for every i. And there're $\frac{n(n-1)}{2}$ inner products $\langle A\_i|A\_j\rangle$. But they're not mutually independent, for instance, if $|\langle A\_1|A\_2\rangle|=|\langle A\_1|A\_3\rangle|=1$, then it's required $|\langle A\_2|A\_3\rangle|=1$. Then my question is what is the exact relations between these inner products? Or how can we compute this relation efficiently?



Relations between a set of inner products of vectors
The matrix of inner products is the Gram matrix of your vectors. This is a positive semi-definite matrix, whose rank is the dimension of the span of the vectors. Conversely, every PSD matrix of the right rank arises as the Gram matrix (since your vectors are normalized, the diagonal elements are all $1$).


