A new information metric and a possible higher bound for a class of measurements in the Quantum Estimation Problem

TL;DR

This paper introduces a new quantum information metric based on phase-space correspondence that can provide higher bounds for parameter estimation in quantum systems, specifically demonstrated for one qubit.

Contribution

It defines a novel class of logarithmic derivatives and shows these can yield tighter bounds than traditional symmetric logarithmic derivatives in quantum estimation.

Findings

Existence of POVMs with higher bounds using the new metric

The new metric surpasses symmetric derivative bounds in one qubit systems

Quantifies deviation from symmetric derivatives via a phase-space function

Abstract

Information metrics give lower bounds for the estimation of parameters. The Cencov-Morozova-Petz Theorem classifies the monotone quantum Fisher metrics. The optimum bound for the quantum estimation problem is offered by the metric which is obtained from the symmetric logarithmic derivative. To get a better bound, it means to go outside this family of metrics, and thus inevitably, to relax some general conditions. In the paper we defined logarithmic derivatives through a phase-space correspondence. This introduces a function which quantifies the deviation from the symmetric derivative. Using this function we have proved that there exist POVMs for which the new metric gives a higher bound from that of the symmetric derivative. The analysis was performed for the one qubit case.