Bayesian Logarithmic Derivative Type Lower Bounds for Quantum Estimation

TL;DR

This paper develops new Bayesian logarithmic derivative type lower bounds for quantum estimation, extending existing bounds and providing a closed-form family of bounds that generalize previous results.

Contribution

It introduces a one-parameter family of Bayesian lower bounds in quantum estimation, generalizing and unifying previous bounds with a closed-form expression.

Findings

Derived a family of Bayesian logarithmic derivative bounds

Unified previous Bayesian lower bounds as special cases

Provided closed-form expressions for the bounds

Abstract

Bayesian approach for quantum parameter estimation has gained a renewed interest from practical applications of quantum estimation theory. Recently, a lower bound, called the Bayesian Nagaoka-Hayashi bound for the Bayes risk in quantum domain was proposed, which is an extension of a new approach to point estimation of quantum states by Conlon et al. (2021). The objective of this paper is to explore this Bayesian Nagaoka-Hayashi bound further by obtaining its lower bounds. We first obtain one-parameter family of lower bounds, which is an analogue of the Holevo bound in point estimation. Thereby, we derive one-parameter family of Bayesian logarithmic derivative type lower bounds in a closed form for the parameter independent weight matrix setting. This new bound includes previously known Bayesian lower bounds as special cases.