In the finite dimensional Hilbert space of quantum mechanics (one where all vectors have norm one), is a concentration of measure phenomenon observed with the inner product of any two vectors? That is, for any $x,y\in\mathcal{H}$, is there an expression of the form
$P(|<x,y> - \gamma| >\epsilon) < f(\epsilon) $
for some $\gamma$ and $f$?
It is my intuition that $\gamma = 1/\sqrt{n}$. I'm rather new to concentration of measure though, so I don't see how to support this claim. Would it be simpler (and still convincing) to heuristically postulate this as a consequence of the law of large numbers?