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

Abstract

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 vectors in $F^n$, $F=\mathbb R$ 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 the complex case which suggests nice patterns on the minimum values. In some cases($p=2$ and partially $n=2$) we supply analytical proofs and give full descriptions of the minimal configurations. We also show that as $m\to \infty$, nearly equidistributed configurations points nearly give the minimal values we expect from our patterns.