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

Understand

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.