Efficient Estimation of Multiple Temperatures via a Collisional Model

TL;DR

This work tackles multi-bath temperature estimation in a collisional quantum thermometry framework by embedding a multiparameter estimation strategy into a sequence of probe–ancilla collisions. A key contribution is a controlled ancilla rotation between collision steps that removes inter-parameter dependencies, rendering the QFIM non-singular and enabling finite precision bounds in the joint estimation of temperatures $T_i$. The authors introduce two benchmarking metrics, $\eta_{\mathrm{joint}}$ against the thermal Fisher information and $\eta_{\mathrm{acc}}$ for accumulated accuracy, and show that joint information can exceed diagonal thermal limits even with uncorrelated ancillas; correlations among ancillas and higher-dimensional ancillas further enhance performance, especially for larger $N$. They provide an experimentally feasible framework in cavity QED and circuit QED to implement the protocol, highlighting a scalable path to high-precision, multi-parameter quantum thermometry by tuning ancilla dimensions and interaction sequences.

Abstract

We present a quantum thermometric protocol for the estimation of multiple temperatures within the collisional model framework. Employing the formalism of multiparameter quantum metrology, we develop a systematic strategy to estimate the temperatures of several thermal reservoirs with minimal estimation error. We prove a necessary and sufficient condition for the singularity of the Fisher information matrix for a bi-parametrized qubit state. By using controlled rotations of ancillary systems between successive interaction stages, we eliminate parameter interdependencies, thereby rendering the quantum Fisher information matrix non-singular. Remarkably, we demonstrate that precision enhancement in the joint estimation of multiple temperatures can be achieved even in the absence of correlations among the ancillas, surpassing the corresponding thermal Fisher information limits. Exploiting correlations within the ancillary system yields additional enhancement of Fisher information. Finally, we identify the dimensionality of the ancillary systems as a key factor governing the efficiency of multiparameter temperature estimation.

Figures (5)

• Figure 1: Schematic representation of the multiparameter temperature estimation protocol based on the collisional model. The information about the first bath, $B_{1}$, is acquired by the $n$-th ancilla through its interaction with the probe system $S_{1}$. A controlled unitary rotation is subsequently applied to the ancilla $A_{n}$, after which it interacts with the next probe to extract information about the second bath, $B_{2}$. This sequence is iteratively repeated for all $N$ thermal reservoirs.
• Figure 2: The variation of $\eta_{\mathrm{acc}}$ is illustrated for the case where a single ancilla sequentially interacts with the first and second probes, with interaction strengths $gt_{S_{1}A_{n}}/\pi$ and $gt_{S_{2}A_{n}}/\pi$, respectively. Here, $K_{B}T_{B_{1}}/\hbar\omega = 2$, and $K_{B}T_{B_{2}}/\hbar\omega = 1$.
• Figure 3: The trend of $\eta_{\mathrm{joint}}$ and $\eta_{\mathrm{acc}}$ with respect to coupling strength for different numbers of uncorrelated ancillas $n$ ($(a)$ and $(c)$). Here $gt_{S_{1}A_{n}}/\pi = 0.5$ and $\gamma t_{S_{i}B_{i}} = 0.5$ for $i = 1,2$. $(b)$ and $(d)$ denotes the scaling of $\eta_{\mathrm{joint}}$, and $\eta_{\mathrm{acc}}$ for different angles of rotation around the $x$-axis. The dotted horizontal lines in $(b)$ and $(d)$ indicate $\eta_{\mathrm{joint}} = 1$ and $\eta_{\mathrm{acc}} = 0$ respectively i.e., $\mathcal{F}_{Q} = \mathcal{F}_{th}$. All other details are the same as Fig. \ref{['2baths_one_anc']}.
• Figure 4: Comparison between correlated (solid lines) and uncorrelated ancilla (dotted lines) for (a) $\eta_{\mathrm{joint}}$ and (b) $\eta_{\mathrm{acc}}$ with respect to the interaction strength. Here, $g t_{S_{1}A_{n}} = 0.5$. All other details are similar to Fig. \ref{['2baths_multi_anc']}.
• Figure 5: The behaviour of (a) $\eta_{\mathrm{joint}}$ and (b) $\eta_{\mathrm{acc}}$ with respect to the interaction strength of the qutrit with the last probe. Here, $g t_{S_{1}A_{n}} = 0.5$, $g t_{S_{2}A_{n}} = 0.2$ and $K_{B}T_{B_{3}}/\hbar\omega = 3$. All other details are similar to Fig. \ref{['2baths_multi_anc']}.

Theorems & Definitions (2)

• Theorem 1
• proof