QestOptPOVM: An iterative algorithm to find optimal measurements for quantum parameter estimation

TL;DR

This paper introduces QestOptPOVM, an iterative steepest descent algorithm that efficiently finds optimal measurements for quantum parameter estimation, improving understanding and validation of bounds in finite-sample quantum metrology.

Contribution

The paper presents a novel algorithm for directly identifying optimal POVMs in quantum estimation, demonstrating its effectiveness through multiple qubit state examples.

Findings

Algorithm efficiently finds optimal measurements for quantum states.

Validation of the Nagaoka-Hayashi bound's tightness in finite samples.

Provides explicit forms of optimal POVMs for quantum estimation.

Abstract

Quantum parameter estimation holds significant promise for achieving high precision through the utilization of the most informative measurements. While various lower bounds have been developed to assess the best accuracy for estimates, they are not tight, nor provide a construction of the optimal measurement in general. Thus, determining the explicit forms of optimal measurements has been challenging due to the non-trivial optimization. In this study, we introduce an algorithm, termed QestOptPOVM, designed to directly identify optimal positive operator-valued measure (POVM) using the steepest descent method. Through rigorous testing on several examples for multiple copies of qubit states (up to six copies), we demonstrate the efficiency and accuracy of our proposed algorithm. Moreover, a comparative analysis between numerical results and established lower bounds serves to validate the tightness of the Nagaoka-Hayashi bound in finite-sample quantum metrology for our examples. Concurrently, our algorithm functions as a tool for elucidating the explicit forms of optimal POVMs, thereby enhancing our understanding of quantum parameter estimation methodologies.