= Maximally entangled overlap bound from Schmidt rank
{title2=$F(|\Phi_d\rangle\langle\Phi_d|,|\zeta\rangle\langle\zeta|)\leq\sqrt{r/d}$}
A normalized bipartite <pure state> of <Schmidt rank> $r$ has <quantum fidelity> at most $\sqrt{r/d}$ with a fixed <maximally entangled state> on $d\otimes d$. Its coefficient <matrix> has <matrix rank> $r$ and <Hilbert-Schmidt norm> one. Its overlap is the <trace> of that <matrix> divided by $\sqrt d$; <trace duality> and the <Cauchy-Schwarz inequality> bound this by $\sqrt{r/d}$. Uniform <Schmidt coefficients> on $r$ matching basis pairs attain equality.
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