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We study the problem of minimizing the energy function $M^p(m,n) := \min\sum_{1\le i<j\le m} |\langle v_i, v_j\rangle|^p$, where $v_i$ are unit vectorsin $F^n$, $F=\mathbb R$ or $\mathbb C$, $m,n,p>0$ are integers and $p$ is even.This problem has implications on finding nice polyhedra in projective spaces,and on quantum random access codes. We conduct experimental search in thecomplex case which suggests nice patterns on the minimum values. In somecases($p=2$ and partially $n=2$) we supply analytical proofs and give fulldescriptions of the minimal configurations. We also show that as $m\to \infty$,nearly equidistributed configurations points nearly give the minimal values weexpect from our patterns.

Energy Minimization in $CP^n$: Some Numerical and Analytical Results