Relations between convergence rates in Schatten p-norms

TL;DR

This paper investigates the relationship between convergence rates in Schatten p-norms and the trace class norm within quantum estimation and tomography, providing insights into how different topologies affect quantum state estimation accuracy.

Contribution

It establishes mathematical relations between convergence rates in Schatten p-norms and the trace class norm, clarifying their interplay in quantum state estimation.

Findings

Derived bounds linking Schatten p-norm and trace class norm convergence rates

Provided theoretical insights into quantum state sampling accuracy

Enhanced understanding of topological differences in quantum state spaces

Abstract

In quantum estimation theory and quantum tomography, the quantum state obtained by sampling converges to the `true' unknown density matrix under topologies that are different from the natural notion of distance in the space of quantum states, i.e. the trace class norm. In this paper, we address such problem, finding relations between the rates of convergence in the Schatten $p$-norms and in the trace class norm.