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

TL;DR

This paper investigates the energy minimization problem for vectors in complex and real spaces, providing numerical experiments, analytical solutions in specific cases, and insights into asymptotic configurations relevant to geometry and quantum information.

Contribution

It offers new analytical solutions for certain cases and numerical evidence for patterns in energy minimization in complex projective spaces.

Findings

Numerical experiments suggest patterns in minimal energy configurations.

Analytical solutions are provided for specific cases ($p=2$, $n=2$).

Nearly equidistributed configurations approach minimal energy as $m$ increases.

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.