# Bayesian Nagaoka-Hayashi Bound for Multiparameter Quantum-State Estimation Problem

## TL;DR

This paper introduces a Bayesian generalization of the Nagaoka-Hayashi bound for quantum state estimation, providing a tighter lower bound than existing bounds and demonstrating efficient computation via semidefinite programming.

## Contribution

It develops a Bayesian version of the Nagaoka-Hayashi bound for multiparameter quantum estimation, extending previous point estimation bounds to Bayesian scenarios.

## Key findings

- The Bayesian Nagaoka-Hayashi bound can be computed efficiently as a semidefinite program.
- The new bound is tighter than Bayesian quantum Cramer-Rao bounds.
- A Bayesian Holevo-type bound is derived from the proposed lower bound.

## Abstract

In this work we propose a Bayesian version of the Nagaoka-Hayashi bound when estimating a parametric family of quantum states. This lower bound is a generalization of a recently proposed bound for point estimation to Bayesian estimation. We then show that the proposed lower bound can be efficiently computed as a semidefinite programming problem. As a lower bound, we also derive a Bayesian version of the Holevo-type bound from the Bayesian Nagaoka-Hayashi bound. Lastly, we prove that the new lower bound is tighter than the Bayesian quantum Cramer-Rao bounds.