Multiparameter quantum estimation with a uniformly accelerated Unruh-DeWitt detector

TL;DR

This work investigates multiparameter quantum estimation for a uniformly accelerated Unruh–DeWitt detector interacting with a vacuum scalar field in Minkowski space, considering both unbounded and bounded vacua. The authors show that the quantum Cramér–Rao bound is not tight for estimating the two parameters $\theta$ (weight) and $\phi$ (phase), and they compute tighter bounds—the Holevo Cramér–Rao bound and the Nagaoka bound—via semidefinite programming, finding the Nagaoka bound to be the tightest among those considered. In the boundary case, the presence of a boundary systematically lowers the bounds, indicating improved estimation precision. The results provide practical guidance for reaching optimal multiparameter precision in relativistic quantum metrology, including insights into when tighter bounds are needed beyond the SLD-CRB or RLD-CRB.

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 Cramér-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 Cramér-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 Cramér-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 Cramér-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.

Figures (4)

• Figure 1: Error bounds as a function of (a) the inverse of acceleration $a$ with $\theta = \pi/2$, $\tau = 0.4$, and $z = 0.5$; (b) the proper time $\tau$ with $\theta = \pi/2$, $a = 0.2$ and $z = 0.5$; (c) the weight parameter $\theta$ with $\tau = 0.4$, $a = 0.2$ and $z = 0.5$.
• Figure 2: Error bounds as a function of (a) the inverse of acceleration $a$ with $\theta = \pi/2$ and $\tau = 0.4$, (b) the proper time $\tau$ with $\theta = \pi/2$ and $a = 0.2$, (c) the weight parameter $\theta$ with $\tau = 0.4$ and $a = 0.2$.
• Figure 3: Error bounds as a function of (a) the inverse of acceleration $a$ with $\theta = \pi/2$, $\tau = 1$ and $z = 0.5$, (b) the proper time $\tau$ with $\theta = \pi/2$, $a = 1$ and $z = 0.5$, (c) the weight parameter $\theta$ with $\tau = 1$, $a = 1$ and $z = 0.5$.
• Figure 4: Error bounds as a function of (a) the inverse of acceleration $a$ with $\theta = \pi/2$, $\tau = 0.4$ and $z = 0.5$, (b) the proper time $\tau$ with $\theta = \pi/2$, $a = 0.2$ and $z = 0.5$, (c) the weight parameter $\theta$ with $\tau = 0.4$, $a = 0.2$ and $z = 0.5$.