Multiparameter quantum estimation with a uniformly accelerated Unruh-DeWitt detector

TL;DR

This paper explores multiparameter quantum estimation using an accelerated Unruh-DeWitt detector, revealing the limitations of traditional bounds and demonstrating improved precision with boundary effects in relativistic quantum metrology.

Contribution

It introduces numerical computation of tighter error bounds for multiparameter estimation in a relativistic setting, highlighting the Nagaoka bound's superiority and boundary effects on estimation precision.

Findings

Nagaoka bound provides the tightest error bound among considered bounds.

Boundary presence reduces estimation error bounds, improving precision.

Quantum Cramér-Rao bound is not tight for two-parameter estimation.

Abstract

The uniformly accelerated Unruh-DeWitt detector serves as a fundamental model in relativistic quantum metrology. While previous studies have mainly concentrated on single-parameter estimation via quantum Cram\'er-Rao bound, the multi-parameter case remains significantly underexplored. In this paper, we investigate the multiparameter estimation for a uniformly accelerated Unruh-DeWitt detector coupled to a vacuum scalar field in both bounded and unbounded Minkowski vacuum. Our analysis reveals that quantum Cram\'er-Rao bound fails to provide a tight error bound for the two-parameter estimation involving the initial phase and weight parameters. For this reason, we numerically compute two tighter error bounds, Holevo Cram\'er-Rao bound and Nagaoka bound, based on a semidefinite program. Notably, our results demonstrate that Nagaoka bound yields the tightest error bound among all the considered error bounds, consistent with the general hierarchy of multiparameter quantum estimation. In the case with a boundary, we observe the introduction of boundary systematically reduces the values of both Holevo Cram\'er-Rao bound and Nagaoka bound, indicating an improvement on the attainable estimation precision. These results offer valuable insights on and practical guidance for advancing multiparameter estimation in relativistic context.