Optimal reconstruction systems for erasures and for the q-potential

TL;DR

This paper introduces the $q$-potential as a new measure for reconstruction systems, extending previous work, and characterizes optimal protocols for data erasures using spectral conditions and inequalities.

Contribution

It extends the Benedetto-Fickus frame potential to a $q$-potential and provides spectral conditions for the existence of optimal reconstruction protocols.

Findings

Protocols minimize the $q$-potential under certain restrictions.

Necessary and sufficient spectral conditions for protocol existence are established.

The work relates reconstruction problems to Klyachko's and Fulton's theory on hermitian operators.

Abstract

We introduce the $q$-potential as an extension of the Benedetto-Fickus frame potential, defined on general reconstruction systems and we show that protocols are the minimizers of this potential under certain restrictions. We extend recent results of B.G. Bodmann on the structure of optimal protocols with respect to 1 and 2 lost packets where the worst (normalized) reconstruction error is computed with respect to a compatible unitarily invariant norm. We finally describe necessary and sufficient (spectral) conditions, that we call $q$-fundamental inequalities, for the existence of protocols with prescribed properties by relating this problem to Klyachko's and Fulton's theory on sums of hermitian operators.