The Frame Potential, on Average

TL;DR

This paper analytically computes the average frame potential for SICs and related structures in quantum information, providing insights into their geometric properties and aiding numerical searches.

Contribution

It derives the average frame potential over Hilbert space and specific subspaces, enhancing understanding of SIC structures and their symmetries.

Findings

Average frame potential computed analytically for SICs

Identifies special subspaces with distinct average properties

Provides tools for improved numerical SIC searches

Abstract

A SIC consists of N^2 equiangular unit vectors in an N dimensional Hilbert space. The frame potential is a function of N^2 unit vectors. It has a unique global minimum if the vectors form a SIC, and this property has been made use of in numerical searches for SICs. When the vectors form an orbit of the Heisenberg group the frame potential becomes a function of a single fiducial vector. We analytically compute the average of this function over Hilbert space. We also compute averages when the fiducial vector is placed in certain special subspaces defined by the Clifford group.